How many questions from Circles appear in JEE?

Circles has appeared in 142 of the last 186 JEE Main and JEE Advanced papers — about 76% of them — contributing 179 questions in total across those papers.

Is Circles an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 76% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Circles questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Circles JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Advanced 2026

Let

P

be the point on the parabola

y = x^2

such that the slope of the tangent to the parabola at the point

P

is 4. Let

Q

be the point in the first quadrant lying on the circle

x^2 + y^2 = 2

such that the slope of the tangent to the circle at the point

Q

is

-1

. Let

R

be the point in the first quadrant lying on the ellipse

x^2 + 4y^2 = 8

such that the slope of the tangent to the ellipse at the point

R

is

-\frac{1}{2}

. Then the radius of the circle passing through the points

P, Q

and

R

is

(A) $\sqrt{10}$
(B) $\sqrt{5}$
(C) $\sqrt{\frac{5}{2}}$
(D) $2\sqrt{5}$

Q2·MathematicsSingle correctJEE Advanced 2026

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
P.The circle with centre $(1, 2)$ and touching the straight line $3x + 4y = 1$ , passes through	1.the point $(1, 1)$
Q.The common tangent to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ with positive slope, passes through	2.the point $(7, 9)$
R.Let $M$ be the end point of the latus rectum of the ellipse $3x^2 + 4y^2 = 48$ such that $M$ lies in the first quadrant. Then the normal to the ellipse drawn at $M$ passes through	3.the point $(3, 2)$
S.Let $H$ be the hyperbola whose centre is at the origin, one of the foci is at $(5, 0)$ , and one directrix is $5x + 16 = 0$ . Then $H$ passes through	4.the point $(2, 5)$
	5.the point $(8, 3\sqrt{3})$

(A)(P) → (3), (Q) → (4), (R) → (1), (S) → (2)
(B)(P) → (3), (Q) → (2), (R) → (1), (S) → (5)
(C)(P) → (3), (Q) → (2), (R) → (4), (S) → (5)
(D)(P) → (4), (Q) → (1), (R) → (2), (S) → (3)

Q3·MathematicsNumericalJEE Main 2026

Consider the circle

C : x^2 + y^2 - 6x - 8y - 11 = 0

. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle

x^2 + y^2 - \alpha x - \beta y - \gamma = 0

, then

\alpha + \beta + 2\gamma

is equal to ________.

Q4·MathematicsNumericalJEE Main 2026

Let the line

x - y = 4

intersect the circle

C : (x - 4)^{2} + (y + 3)^{2} = 9

at the points

Q

and

R

. If

P(\alpha, \beta)

is a point on

C

such that

PQ = PR

, then

(6\alpha + 8\beta)^{2}

is equal to __________.

Q5·MathematicsSingle correctJEE Main 2026

Let

C

be a circle having centre in the first quadrant and touching the

x

-axis at a distance of 3 units from the origin. If the circle

C

has an intercept of length

6\sqrt{3}

on

y

-axis, then the length of the chord of the circle

C

on the line

x - y = 3

is :

(A)8
(B)6
(C) $6\sqrt{2}$
(D) $8\sqrt{2}$

Q6·MathematicsNumericalJEE Main 2026

Let the centre of the circle x

^2

+ y

^2

+ 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2x − y = 4. Let the area of an equilateral triangle inscribed in the circle be

27\sqrt{3}

. Then the square of the length of the chord of the circle on the line x = 1 is ______.

Q7·MathematicsSingle correctJEE Main 2026

Let

P

be a moving point on the circle

x^2 + y^2 - 6x - 8y + 21 = 0

. Then, the maximum distance of

P

from the vertex of the parabola

x^2 + 6x + y + 13 = 0

is equal to:

(A)8
(B)10
(C)12
(D)9

Q8·MathematicsSingle correctJEE Main 2026

Let the point P be the vertex of the parabola

y = x^2 - 6x + 12

. If a line passing through the point P intersects the circle

x^2 + y^2 - 2x - 4y + 3 = 0

at the points R and S, then the maximum value of

(PR + PS)^2

is :

(A)10
(B)20
(C)25
(D)5

Q9·MathematicsSingle correctJEE Main 2026

Suppose that two chords, drawn from the point

(1, 2)

on the circle

x^2 + y^2 + x - 3y = 0

are bisected by the

y

-axis. If the other ends of these chords are

R

and

S

, and the mid point of the line segment

RS

is

(\alpha, \beta)

, then

6(\alpha + \beta)

is equal to:

(A) $1$
(B) $3$
(C) $4$
(D) $6$

Q10·MathematicsSingle correctJEE Main 2026

Let

A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}

and

\det(A - \alpha I) = 0

, where

\alpha

is a real number. If the largest possible value of

\alpha

is

p

, then the circle

(x - p)^{2} + (y - 2p)^{2} = 320

, intersects the co-ordinate axes at

(A)1 point
(B)2 points
(C)3 points
(D)4 points

Q11·MathematicsSingle correctJEE Main 2026

Let

P(3\cos\alpha, 2\sin\alpha)

,

\alpha \neq 0

, be a point on the ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1

,

Q

be a point on the circle

x^{2} + y^{2} - 14x - 14y + 82 = 0

and

R

be a point on the line

x + y = 5

such that the centroid of the triangle

PQR

is

\left(2 + \cos\alpha, 3 + \frac{2}{3}\sin\alpha\right)

. Then the sum of the ordinates of all possible points

R

is:

(A)6
(B)2
(C)4
(D)8

Q12·MathematicsNumericalJEE Main 2026

Let a circle

C

have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of

C

on the line

x + y = 1

is

\sqrt{14}

, then the square of the radius of

C

is ________.

Q13·MathematicsSingle correctJEE Main 2026

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines

x + (k - 1)y + 3 = 0

and

2x + k^2y - 4 = 0

. If the line

x - y + 2 = 0

intersects the circle at the points A and B, then

(AB)^2

is equal to:

(A) $10$
(B) $27$
(C) $18$
(D) $34$

Q14·MathematicsSingle correctJEE Main 2026

Let the circle

x^{2} + y^{2} = 4

intersect x-axis at the points A(a, 0), a > 0 and B(b,0). Let P(2 cosα, 2 sinα),

0 < \alpha < \dfrac{\pi}{2}

and Q(2 cosβ, 2 sinβ) be two points such that

(\alpha - \beta) = \dfrac{\pi}{2}

. Then the point of intersection of AQ and BP lies on :

(A) $x^{2} + y^{2} - 4y - 4 = 0$
(B) $x^{2} + y^{2} - 4x - 4 = 0$
(C) $x^{2} + y^{2} - 4x - 4y = 0$
(D) $x^{2} + y^{2} - 4x - 4y - 4 = 0$

Q15·MathematicsSingle correctJEE Main 2026

Let

y = x

be the equation of a chord of the circle

C_1

(in the closed half-plane

x \ge 0

) of diameter 10 passing through the origin. Let

C_2

be another circle described on the given chord as its diameter. If the equation of the chord of the circle

C_2

, which passes through the point

(2, 3)

and is farthest from the center of

C_2

, is

x + ay + b = 0

, then

a - b

is equal to :

(A)10
(B)−6
(C)−2
(D)6

Q16·MathematicsSingle correctJEE Main 2026

Let a circle of radius 4 pass through the origin O, the points A

\left(-\sqrt{3}a,0\right)

and B

\left(0,-\sqrt{2}\,b\right)

, where a and b are real parameters and ab ≠ 0. Then the locus of the centroid of ΔOAB is a circle of radius

(A) $\frac{5}{3}$
(B) $\frac{7}{3}$
(C) $\frac{8}{3}$
(D) $\frac{11}{3}$

Q17·MathematicsSingle correctJEE Main 2026

Let the set of all values of r, for which the circles

(x + 1)^2 + (y + 4)^2 = r^2

and

x^2 + y^2 - 4x - 2y - 4 = 0

intersect at two distinct points be the interval (α, β). Then αβ is equal to

(A)25
(B)20
(C)21
(D)24

Q18·MathematicsSingle correctJEE Main 2026

Let

\vec{c}

and

\vec{d}

be vectors such that

|\vec{c} + \vec{d}| = \sqrt{29}

and

\vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d}

. If

\lambda_{1}, \lambda_{2}(\lambda_{1} > \lambda_{2})

are the possible values of

(\vec{c} + \vec{d}).(-7\hat{i} + 2\hat{j} + 3\hat{k})

, then the equation

K^{2}x^{2} + (K^{2} - 5K + \lambda_{1})xy + \left(3K + \frac{\lambda_{2}}{2}\right)y^{2} - 8x + 12y + \lambda_{2} = 0

represents a circle, for k equal to :

(A)4
(B)1
(C)–1
(D)2

Q19·MathematicsSingle correctJEE Main 2026

Let PQ and MN be two straight lines touching the circle

x^{2}+y^{2}-4x-6y-3=0

at the points A and B respectively. Let O be the centre of the circle and ∠AOB = π/3. Then the locus of the point of intersection of the lines PQ and MN is:

(A) $3(x^{2}+y^{2})-18x-12y+25=0$
(B) $x^{2}+y^{2}-12x-18y-25=0$
(C) $x^{2}+y^{2}-18x-12y-25=0$
(D) $3(x^{2}+y^{2})-12x-18y-25=0$

Q20·MathematicsNumericalJEE Main 2026

If P is a point on the circle

x^{2} + y^{2} = 4

, Q is a point on the straight line

5x + y + 2 = 0

and

x - y + 1 = 0

is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is _______.

Q21·MathematicsIntegerJEE Main 2025

Let

r

be the radius of the circle, which touches x-axis at point

(a,0)

,

a<0

and the parabola

y^2=9x

at the point

(4,6)

. Then

r

is equal to ___

Q22·MathematicsSingle correctJEE Main 2025

Let the ellipse

3x^2+py^2=4

pass through the centre

C

of the circle

x^2+y^2-2x-4y-11=0

of radius

r

. Let

f_1,f_2

be the focal distances of the point

C

on the ellipse. Then

6f_1f_2-r

is equal to:

(A) $74$
(B) $68$
(C) $70$
(D) $78$

Q23·MathematicsSingle correctJEE Main 2025

If the orthocentre of the triangle formed by the lines

y=x+1

,

y=4x-8

and

y=mx+c

is at

(3,-1)

, then

m-c

is:

(A)0
(B) $-2$
(C)4
(D)2

Q24·MathematicsSingle correctJEE Main 2025

Let

C_1

be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let

C_2

be the circle with centre

(1,3)

that touches

C_1

externally at the point

(\alpha,\beta)

. If

(\beta-\alpha)^2=\dfrac{m}{n}

,

\gcd(m,n)=1

, then

m+n

is equal to:

(A)9
(B)13
(C)22
(D)31

Q25·MathematicsIntegerJEE Main 2025

Let

C

be the circle

x^2+(y-1)^2=2

,

E_1

and

E_2

be two ellipses whose centres lie at the origin and major axes lie on

x

-axis and

y

-axis respectively. Let the straight line

x+y=3

touch the curves

C

,

E_1

and

E_2

at

P(x_1,y_1)

,

Q(x_2,y_2)

and

R(x_3,y_3)

respectively. Given that

P

is the mid-point of the line segment

QR

and

PQ=\dfrac{2\sqrt{2}}{3}

, the value of

9(x_1y_1+x_2y_2+x_3y_3)

is equal to ______.

Q26·MathematicsSingle correctJEE Main 2025

Let the three sides of a triangle are on the lines

4x-7y+10=0

,

x+y=5

and

7x+4y=15

. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines

x=0

,

y=0

and

x+y=1

is

(A)5
(B) $\sqrt{5}$
(C) $\sqrt{20}$
(D)20

Q27·MathematicsSingle correctJEE Main 2025

If the four distinct points

(4,6)

,

(-1,5)

,

(0,0)

and

(k,3k)

lie on a circle of radius

r

, then

10k+r^2

is equal to:

(A)32
(B)33
(C)34
(D)35

Q28·MathematicsSingle correctJEE Main 2025

The radius of the smallest circle which touches the parabolas

y=x^2+2

and

x=y^2+2

is:

(A) $\dfrac{7\sqrt2}{2}$
(B) $\dfrac{7\sqrt2}{16}$
(C) $\dfrac{7\sqrt2}{4}$
(D) $\dfrac{7\sqrt2}{8}$

Q29·MathematicsIntegerJEE Main 2025

The absolute difference between the squares of the radii of the two circles passing through the point

(-9,4)

and touching the lines

x+y=3

and

x-y=3

, is equal to ______.

Q30·MathematicsSingle correctJEE Main 2025

Let the line

x+y=1

meet the circle

x^2+y^2=4

at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to

(A) $3\sqrt{7}$
(B) $2\sqrt{14}$
(C) $5\sqrt{7}$
(D) $\sqrt{14}$

Q31·MathematicsSingle correctJEE Main 2025

Let a circle C pass through the points

(4, 2)

and

(0, 2)

, and its centre lie on

3x+2y+2=0

. Then the length of the chord, of the circle C, whose mid-point is

(1, 2)

, is:

(A) $\sqrt{3}$
(B) $2\sqrt{3}$
(C) $4\sqrt{2}$
(D) $2\sqrt{2}$

Q32·MathematicsSingle correctJEE Main 2025

Let the equation of the circle, which touches x-axis at the point

(a, 0)

,

a>0

and cuts off an intercept of length b on y-axis be

x^2+y^2-\alpha x-\beta y+\gamma=0

. If the circle lies below x-axis, then the ordered pair

(2a, b^2)

is equal to:

(A) $(\alpha, \beta^2+4\gamma)$
(B) $(\gamma, \beta^2-4\alpha)$
(C) $(\gamma, \beta^2+4\alpha)$
(D) $(\alpha, \beta^2-4\gamma)$

Q33·MathematicsSingle correctJEE Main 2025

Let circle C be the image of

x^2+y^2-2x+4y-4=0

in the line

2x-3y+5=0

and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C. If

B(\alpha,\beta)

, with

\beta<4

, lies on C such that the length of the arc AB is

\dfrac{1}{6}

th of the perimeter of C, then

\beta-\sqrt3\alpha

is equal to

(A) $3$
(B) $3+\sqrt3$
(C) $4-\sqrt3$
(D) $4$

Q34·MathematicsSingle correctJEE Main 2025

Let the shortest distance from

(a,0),\ a>0,

to the parabola

y^2=4x

be

4

. Then the equation of the circle passing through the point

(a,0)

and the focus of the parabola, and having its centre on the axis of the parabola is :

(A) $x^2+y^2-6x+5=0$
(B) $x^2+y^2-4x+3=0$
(C) $x^2+y^2-10x+9=0$
(D) $x^2+y^2-8x+7=0$

Q35·MathematicsIntegerJEE Main 2025

The focus of the parabola

y^2=4x+16

is the centre of the circle

C

of radius

5

. If the values of

\lambda

, for which

C

passes through the point of intersection of the lines

3x-y=0

and

x+\lambda y=4

, are

\lambda_1

and

\lambda_2,\ \lambda_1<\lambda_2

, then

12\lambda_1+29\lambda_2

is equal to __________.

Q36·MathematicsIntegerJEE Main 2025

Let the circle

C

touch the line

x-y+1=0

, have the centre on the positive x-axis, and cut off a chord of length

\dfrac{4}{\sqrt{13}}

along the line

-3x+2y=1

. Let

H

be the hyperbola

\dfrac{x^2}{\alpha^2}-\dfrac{y^2}{\beta^2}=1

, whose one of the foci is the centre of

C

and the length of the transverse axis is the diameter of

C

. Then

2\alpha^2+3\beta^2

is equal to _______

Q37·MathematicsSingle correctJEE Main 2025

Let the parabola

y=x^2+px-3

meet the axes at P, Q, R. If the circle with centre

(-1,-1)

passes through P, Q, R, then the area of

\triangle PQR

is:

(A)4
(B)6
(C)7
(D)5

Q38·MathematicsSingle correctJEE Main 2025

A circle C of radius 2 lies in the second quadrant and touches both axes. Another circle has centre

(2,5)

and radius

r

and intersects C at exactly two points. If the set of possible

r

is

(\alpha,\beta)

, then

3\beta-2\alpha

equals:

(A)15
(B)24
(C)12
(D)10

Q39·MathematicsSingle correctJEE Advanced 2024

Let the straight line

y = 2x

touch a circle with centre

(0, \alpha)

,

\alpha > 0

, and radius

r

at a point

A_{1}

. Let

B_{1}

be the point on the circle such the line segment

A_{1}B_{1}

is a diameter of the circle. Let

\alpha + r = 5 + \sqrt{5}

. Match each entry in List-I to the correct entries in List-II. The correct option is:

List-I	List-II
P. $\alpha$ equals	1. $(-2, 4)$
Q. $r$ equals	2. $\sqrt{5}$
R. $A_{1}$ equals	3. $(-2, 6)$
S. $B_{1}$ equals	4.5
	5. $(2, 4)$

(A)(P) $\to$ (4) (Q) $\to$ (2) (R) $\to$ (1) (S) $\to$ (3)
(B)(P) $\to$ (2) (Q) $\to$ (4) (R) $\to$ (1) (S) $\to$ (3)
(C)(P) $\to$ (4) (Q) $\to$ (2) (R) $\to$ (5) (S) $\to$ (3)
(D)(P) $\to$ (2) (Q) $\to$ (4) (R) $\to$ (3) (S) $\to$ (5)

Q40·MathematicsNumericalJEE Main 2024

Let the centre of a circle, passing through the point

(0, 0)

,

(1, 0)

and touching the circle

x^2 + y^2 = 9

, be

(h, k)

. Then for all possible values of the coordinates of the centre

(h, k)

,

4(h^2 + k^2)

is equal to ________.

Q41·MathematicsSingle correctJEE Main 2024

Let a circle passing through

(2, 0)

have its centre at the point

(h, k)

. Let

(x_c, y_c)

be the point of intersection of the lines

3x + 5y = 1

and

(2 + c)x + 5c^2 y = 1

. If

h = \lim_{c \to 1} x_c

and

k = \lim_{c \to 1} y_c

, then the equation of the circle is:

(A) $25x^2 + 25y^2 - 20x + 2y - 60 = 0$
(B) $5x^2 + 5y^2 - 4x - 2y - 12 = 0$
(C) $25x^2 + 25y^2 - 2x + 2y - 60 = 0$
(D) $5x^2 + 5y^2 - 4x + 2y - 12 = 0$

Q42·MathematicsSingle correctJEE Main 2024

Let the circles

C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2

and

C_2:(x-8)^2+\left(y-\dfrac{15}{2}\right)^2=r_2^2

touch each other externally at the point

(6,6)

. If the point

(6,6)

divides the line segment joining the centres of the circles

C_1

and

C_2

internally in the ratio

2:1

, then

(\alpha+\beta)+4(r_1^2+r_2^2)

equals:

(A) $110$
(B) $130$
(C) $125$
(D) $145$

Q43·MathematicsSingle correctJEE Main 2024

If the locus of the point, whose distances from the point

(2, 1)

and

(1, 3)

are in the ratio

5 : 4

, is

ax^2 + by^2 + cxy + dx + ey + 170 = 0

, then the value of

a^2 + 2b + 3c + 4d + e

is equal to:

(A)5
(B) $-27$
(C)37
(D)437

Q44·MathematicsSingle correctJEE Main 2024

Let

C

be the circle of minimum area touching the parabola

y=6-x^{2}

and the lines

y=\sqrt{3}\,|x|

. Then, which one of the following points lies on the circle

C

?

(A) $(2,4)$
(B) $(1,2)$
(C) $(2,2)$
(D) $(1,1)$

Q45·MathematicsSingle correctJEE Main 2024

Let the circle

C_1:x^2+y^2-2(x+y)+1=0

and

C_2

be a circle having centre at

(-1,0)

and radius 2. If the line of the common chord of

C_1

and

C_2

intersects the y-axis at the point

P

, then the square of the distance of

P

from the centre of

C_1

is:

(A)2
(B)1
(C)6
(D)4

Q46·MathematicsSingle correctJEE Main 2024

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point

(3, 2)

and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point

(5, 5)

is:

(A) $2\sqrt{2}$
(B) $5$
(C) $4\sqrt{2}$
(D) $4$

Q47·MathematicsNumericalJEE Main 2024

Let the maximum and minimum values of

\left(\sqrt{8x-x^2-12}-4\right)^2+(x-7)^2

,

x\in\mathbb{R}

, be

M

and

m

respectively. Then

M^2-m^2

is equal to __________.

Q48·MathematicsSingle correctJEE Main 2024

A square is inscribed in the circle

x^2+y^2-10x-6y+30=0

. One side of this square is parallel to

y=x+3

. If

(x_i,y_i)

are the vertices of the square, then

\sum(x_i^2+y_i^2)

is equal to:

(A) $148$
(B) $156$
(C) $160$
(D) $152$

Q49·MathematicsSingle correctJEE Main 2024

Let C be a circle with radius

\sqrt{10}

units and centre at the origin. Let the line

x+y=2

intersect the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope

-1

. Then, a distance (in units) between the chord PQ and the chord MN is

(A) $2-\sqrt{3}$
(B) $3-\sqrt{2}$
(C) $\sqrt{2}-1$
(D) $\sqrt{2}+1$

Q50·MathematicsSingle correctJEE Main 2024

Let the locus of the mid points of the chords of circle

x^2+(y-1)^2=1

drawn from the origin intersect the line

x+y=1

at P and Q. Then, the length of PQ is:

(A) $\dfrac{1}{\sqrt 2}$
(B) $\sqrt 2$
(C) $\dfrac{1}{2}$
(D)1

Q51·MathematicsSingle correctJEE Main 2024

Let

C:x^2+y^2=4

and

C':x^2+y^2-4\lambda x+9=0

be two circles. If the set of all values of

\lambda

so that the circles

C

and

C'

intersect at two distinct points, is

\mathbb{R}-[a,b]

, then the point

(8a+12,16b-20)

lies on the curve:

(A) $x^2+2y^2-5x+6y=3$
(B) $5x^2-y=-11$
(C) $x^2-4y^2=7$
(D) $6x^2+y^2=42$

Q52·MathematicsNumericalJEE Main 2024

Let the line

\sqrt2\,x+y=\alpha

pass through the point of the intersection

P

(in the first quadrant) of the circle

x^2+y^2=3

and the parabola

x^2=2y

. Let the line

L

touch two circles

C_1

and

C_2

of equal radius

2\sqrt3

. If the centres

Q_1

and

Q_2

of the circles

C_1

and

C_2

lie on the y-axis, then the square of the area of the triangle

PQ_1Q_2

is equal to ___

Q53·MathematicsSingle correctJEE Main 2024

Let a variable line passing through the centre of the circle

x^2+y^2-16x-4y=0

, meet the positive co-ordinate axes at the points A and B. Then the minimum value of

OA+OB

, where O is the origin, is equal to

(A)12
(B)18
(C)20
(D)24

Q54·MathematicsSingle correctJEE Main 2024

If one of the diameters of the circle

x^2+y^2-10x+4y+13=0

is a chord of another circle C, whose center is the point of intersection of the lines

2x+3y=12

and

3x-2y=5

, then the radius of the circle C is

(A) $\sqrt{20}$
(B) $4$
(C) $6$
(D) $3\sqrt{2}$

Q55·MathematicsNumericalJEE Main 2024

Consider two circles

C_1:x^2+y^2=25

and

C_2:(x-\alpha)^2+y^2=16

. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of

C_1

and

C_2

be

\sin^{-1}\left(\dfrac{\sqrt{63}}{8}\right)

. If the length of common chord of

C_1

and

C_2

is

\beta

, then the value of

(\alpha\beta)^2

equals ______.

Q56·MathematicsSingle correctJEE Main 2024

If the circles

(x+1)^2+(y+2)^2=r^2

and

x^2+y^2-4x-4y+4=0

intersect at exactly two distinct points, then:

(A) $5<r<7$
(B) $0<r<7$
(C) $3<r<7$
(D) $\dfrac12<r<7$

Q57·MathematicsSingle correctJEE Main 2024

Let

\left(5,\dfrac{a}{4}\right)

be the circumcenter of a triangle with vertices

A(a,-2)

,

B(a,6)

and

C\left(\dfrac{a}{4},-2\right)

. Let

\alpha

denote the circumradius,

\beta

denote the area and

\gamma

denote the perimeter of the triangle. Then

\alpha+\beta+\gamma

is

(A) $60$
(B) $53$
(C) $62$
(D) $30$

Q58·MathematicsNumericalJEE Main 2024

Equation of two diameters of a circle are

2x-3y=5

and

3x-4y=7

. The line joining the points

\left(-\dfrac{22}{7},-4\right)

and

\left(-\dfrac{1}{7},3\right)

intersects the circle at only one point

P(\alpha,\beta)

. Then

17\beta-\alpha

is equal to ______.

Q59·MathematicsSingle correctJEE Main 2024

If the shortest distance of the parabola

y^2=4x

from the centre of the circle

x^2+y^2-4x-16y+64=0

is

d

, then

d^2

is equal to:

(A)16
(B)24
(C)20
(D)36

Q60·MathematicsSingle correctJEE Main 2024

Four distinct points

(2k,3k)

,

(1,0)

,

(0,1)

and

(0,0)

lie on a circle for

k

equal to:

(A) $\dfrac{2}{13}$
(B) $\dfrac{3}{13}$
(C) $\dfrac{5}{13}$
(D) $\dfrac{1}{13}$

Q61·MathematicsNumericalJEE Main 2024

Consider a circle

(x-\alpha)^2+(y-\beta)^2=50

, where

\alpha,\beta>0

. If the circle touches the line

y+x=0

at the point P, whose distance from the origin is

4\sqrt2

, then

(\alpha+\beta)^2

is equal to __________.

Q62·MathematicsIntegerJEE Advanced 2023

Let

C_{1}

be the circle of radius 1 with center at the origin. Let

C_{2}

be the circle of radius

r

with center at the point

A = (4, 1)

, where

1 < r < 3

. Two distinct common tangents

PQ

and

ST

of

C_{1}

and

C_{2}

are drawn. The tangent

PQ

touches

C_{1}

at

P

and

C_{2}

at

Q

. The tangent

ST

touches

C_{1}

at

S

and

C_{2}

at

T

. Mid points of the line segments

PQ

and

ST

are joined to form a line which meets the x-axis at a point

B

. If

AB = \sqrt{5}

, then the value of

r^{2}

is

Q63·MathematicsSingle correctJEE Main 2023

The number of common tangents, to the circles

x^2+y^2-18x-15y+131=0

and

x^2+y^2-6x-6y-7=0

, is :

(A)3
(B)2
(C)1
(D)4

Q64·MathematicsSingle correctJEE Main 2023

Let the centre of a circle C be

(\alpha, \beta)

and its radius

r < 8

. Let

3x + 4y = 24

and

3x - 4y = 32

be two tangents and

4x + 3y = 1

be a normal to C. Then

(\alpha - \beta + r)

is equal to

(A)7
(B)9
(C)5
(D)6

Q65·MathematicsNumericalJEE Main 2023

Two circles in the first quadrant of radii

r_1

and

r_2

touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line

x+y=2

. Then

r_1^2+r_2^2-r_1 r_2

is equal to _________.

Q66·MathematicsSingle correctJEE Main 2023

A line segment AB of length

\lambda

moves such that the points A and B remain on the periphery of a circle of radius

\lambda

. Then the locus of the point, that divides the line segment AB in the ratio

2:3

, is a circle of radius

(A) $\dfrac35\lambda$
(B) $\dfrac{\sqrt{19}}{7}\lambda$
(C) $\dfrac23\lambda$
(D) $\dfrac{\sqrt{19}}{5}\lambda$

Q67·MathematicsSingle correctJEE Main 2023

Let A be the point

(1,2)

and B be any point on the curve

x^2+y^2=16

. If the centre of the locus of the point P, which divides the line segment AB in the ratio

3:2

, is the point

C(\alpha,\beta)

, then the length of the line segment AC is:

(A) $\frac{6\sqrt{5}}{5}$
(B) $\frac{4\sqrt{5}}{5}$
(C) $\frac{2\sqrt{5}}{5}$
(D) $\frac{3\sqrt{5}}{5}$

Q68·MathematicsSingle correctJEE Main 2023

Let O be the origin and OP and OQ be the tangents to the circle

x^{2}+y^{2}-6x+4y+8=0

at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point

\left(\alpha,\dfrac{1}{2}\right)

, then a value of

\alpha

is

(A) $\dfrac{3}{2}$
(B) $\dfrac{5}{2}$
(C) $1$
(D) $-\dfrac{1}{2}$

Q69·MathematicsNumericalJEE Main 2023

Consider a circle

C_{1}:x^{2}+y^{2}-4x-2y=\alpha-5

. Let its mirror image in the line

y=2x+1

be another circle

C_{2}:5x^{2}+5y^{2}-10fx-10gy+36=0

. Let

r

be the radius of

C_{2}

. Then

\alpha+r

is equal to

Q70·MathematicsNumericalJEE Main 2023

A circle passing through the point

P(\alpha,\beta)

in the first quadrant touches the two coordinate axes at the points

A

and

B

. The point

P

is above the line

AB

. The point

Q

on the line segment

AB

is the foot of perpendicular from

P

on

AB

. If

PQ

is equal to

11

units, then the value of

\alpha\beta

is _____.

Q71·MathematicsSingle correctJEE Main 2023

If the tangents at the points

P

and

Q

on the circle

x^2+y^2-2x+y=5

meet at the point

R\left(\frac{9}{4},-\frac{2}{4}\right)

, then the area of the triangle

PQR

is

(A) $\frac{13}{4}$
(B) $\frac{13}{8}$
(C) $\frac{5}{4}$
(D) $\frac{5}{8}$

Q72·MathematicsSingle correctJEE Main 2023

The set of all values of

a^2

for which the line

x+y=0

bisects two distinct chords drawn from a point

P\left(\dfrac{1+a}{2},\dfrac{1-a}{2}\right)

on the circle

2x^2+2y^2-(1+a)x-(1-a)y=0

, is equal to:

(A) $(0,4]$
(B) $(4,\infty)$
(C) $(2,12]$
(D) $(8,\infty)$

Q73·MathematicsSingle correctJEE Main 2023

Let a circle

C_1

be obtained on rolling the circle

x^2+y^2-4x-6y+11=0

upwards 4 units on the tangent

T

to it at the point

(3,2)

. Let

C_2

be the image of

C_1

in

T

. Let

A

and

B

be the centers of circles

C_1

and

C_2

respectively, and

M

and

N

be respectively the feet of perpendiculars drawn from

A

and

B

on the

x

-axis. Then the area of the trapezium AMNB is :

(A) $4(1+\sqrt{2})$
(B) $3+2\sqrt{2}$
(C) $2(1+\sqrt{2})$
(D) $2(2+\sqrt{2})$

Q74·MathematicsSingle correctJEE Main 2023

Let

y=x+2

,

4y=3x+6

and

3y=4x+1

be three tangent lines to the circle

(x-h)^2+(y-k)^2=r^2

. Then

h+k

is equal to:

(A) $5(1+\sqrt2)$
(B) $5\sqrt2$
(C) $6$
(D) $5$

Q75·MathematicsNumericalJEE Main 2023

Let

P(a_1, b_1)

and

Q(a_2, b_2)

be two distinct points on a circle with center

C(\sqrt{2}, \sqrt{3})

. Let

O

be the origin and

OC

be perpendicular to both

CP

and

CQ

. If the area of the triangle

OCP

is

\dfrac{\sqrt{35}}{2}

, then

a_1^2 + a_2^2 + b_1^2 + b_2^2

is equal to _______.

Q76·MathematicsNumericalJEE Main 2023

A circle with centre

(2,3)

and radius

4

intersects the line

x+y=3

at the points

P

and

Q

. If the tangents at

P

and

Q

intersect at the point

S(\alpha,\beta)

, then

4\alpha-7\beta

is equal to _____.

Q77·MathematicsSingle correctJEE Main 2023

Let the tangents at the points

A(4, -11)

and

B(8, -5)

on the circle

x^2 + y^2 - 3x + 10y - 15 = 0

, intersect at the point

C

. Then the radius of the circle, whose centre is

C

and the line joining

A

and

B

is its tangent, is equal to

(A) $2\sqrt{13}$
(B) $\sqrt{13}$
(C) $\dfrac{3\sqrt{3}}{4}$
(D) $\dfrac{2\sqrt{13}}{3}$

Q78·MathematicsNumericalJEE Main 2023

Points

P(-3,2)

,

Q(9,10)

and

R(a,4)

lie on a circle

C

with

PR

as its diameter. The tangents to

C

at the points

Q

and

R

intersect at the point

S

. If

S

lies on the line

2x-ky=1

, then

k

is equal to

Q79·MathematicsSingle correctJEE Main 2023

The points of intersection of the line

ax+by=0

,

(a\ne b)

and the circle

x^2+y^2-2x=0

are

A(\alpha,0)

and

B(1,\beta)

. The image of the circle with

AB

as a diameter in the line

x+y+2=0

is:

(A) $x^2+y^2+3x+3y+4=0$
(B) $x^2+y^2+3x+5y+8=0$
(C) $x^2+y^2-5x-5y+12=0$
(D) $x^2+y^2+5x+5y+12=0$

Q80·MathematicsSingle correctJEE Main 2023

The locus of the mid points of the chords of the circle

C_1:(x-4)^2+(y-5)^2=4

which subtend an angle

\theta_i

at the centre of the circle

C_1

, is a circle of radius

r_i

. If

\theta_1=\dfrac{\pi}{3},\ \theta_3=\dfrac{2\pi}{3}

and

r_1^2=r_2^2+r_3^2

, then

\theta_2

is equal to

(A) $\dfrac{\pi}{4}$
(B) $\dfrac{\pi}{2}$
(C) $\dfrac{\pi}{6}$
(D) $\dfrac{3\pi}{4}$

Q81·MathematicsMultiple correctJEE Advanced 2022

Let G be a circle of radius R > 0. Let G

_{1}

, G

_{2}

, …, G

_{n}

be n circles of equal radius r > 0. Suppose each of the n circles G

_{1}

, G

_{2}

, …, G

_{n}

touches the circle G externally. Also, for i = 1, 2, …, n − 1, the circle G

_{i}

touches G

_{i+1}

externally, and G

_{n}

touches G

_{1}

externally. Then, which of the following statements is/are TRUE?

(A)If n = 4, then $\left(\sqrt{2}-1\right)$ r < R
(B)If n = 5, then r < R
(C)If n = 8, then $\left(\sqrt{2}-1\right)$ r < R
(D)If n = 12, then $\sqrt{2}\left(\sqrt{3}+1\right)$ r > R .

Q82·MathematicsNumericalJEE Main 2022

Let the mirror image of a circle

c_{1} : x^{2}+y^{2}-2x-6y+\alpha = 0

in line

y = x + 1

be

c_{2} : 5x^{2}+5y^{2}+10gx+10fy+38 = 0

. If r is the radius of circle

c_{2}

, then

\alpha + 6r^{2}

is equal to __________

Q83·MathematicsNumericalJEE Main 2022

Let AB be a chord of length 12 of the circle

(x - 2)^2 + (y + 1)^2 = \frac{169}{4}

. If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to ___.

Q84·MathematicsSingle correctJEE Main 2022

Let the tangents at two points A and B on the circle

x^{2}+y^{2}-4x+3=0

meet at origin

O(0,0)

. Then the area of the triangle of OAB is

(A) $\frac{3\sqrt{3}}{2}$
(B) $\frac{3\sqrt{3}}{4}$
(C) $\frac{3}{2\sqrt{3}}$
(D) $\frac{3}{4\sqrt{3}}$

Q85·MathematicsSingle correctJEE Main 2022

Let C be the centre of the circle

x^{2} + y^{2} - x + 2y = \frac{11}{4}

and P be a point on the circle. A line passes through the point C, makes an angle of

\frac{\pi}{4}

with the line CP and intersects the circle at the points Q and R. Then the area of the triangle PQR (in unit

^{2}

) is :

(A)2
(B) $2\sqrt{2}$
(C) $8\sin\left(\frac{\pi}{8}\right)$
(D) $8\cos\left(\frac{\pi}{8}\right)$

Q86·MathematicsSingle correctJEE Main 2022

If the circle

x^{2}+y^{2}-2gx+6y-19c=0

,

g,c\in\mathbb{R}

passes through the point (6, 1) and its centre lies on the line x − 2cy = 8, then the length of intercept made by the circle on x-axis is

(A) $\sqrt{11}$
(B)4
(C)3
(D) $2\sqrt{23}$

Q87·MathematicsSingle correctJEE Main 2022

A circle

C_{1}

passes through the origin O and has diameter 4 on the positive x-axis. The line

y = 2x

gives a chord OA of a circle

C_{1}

. Let

C_{2}

be the circle with OA as a diameter. If the tangent to

C_{2}

at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to :

(A) $1 : 4$
(B) $1 : 5$
(C) $2 : 5$
(D) $1 : 3$

Q88·MathematicsSingle correctJEE Main 2022

A point P moves so that the sum of squares of its distances from the points (1, 2) and (–2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A, B and the y-axis at the point C, D . Then the area of the quadrilateral ACBD is equal to

(A) $\dfrac{9}{2}$
(B) $\dfrac{3\sqrt{17}}{2}$
(C) $\dfrac{3\sqrt{17}}{4}$
(D)9

Q89·MathematicsSingle correctJEE Main 2022

Let the abscissae of the two points P and Q on a circle be the roots of

x^{2} - 4x - 6 = 0

and the ordinates of P and Q be the roots of

y^{2} + 2y - 7 = 0

. If PQ is a diameter of the circle

x^{2} + y^{2} + 2ax + 2by + c= 0

, then the value of

(a+b-c)

is

(A)12
(B)13
(C)14
(D)16

Q90·MathematicsSingle correctJEE Main 2022

Let the locus of the centre

(\alpha, \beta)

,

\beta > 0

, of the circle which touches the circle

x^{2} + (y - 1)^{2} = 1

externally and also touches the x-axis be L. Then the area bounded by L and the line

y = 4

is :

(A) $\frac{32\sqrt{2}}{3}$
(B) $\frac{40\sqrt{2}}{3}$
(C) $\frac{64}{3}$
(D) $\frac{32}{3}$

Q91·MathematicsNumericalJEE Main 2022

The sum of diameters of the circles that touch (i) the parabola

75x^{2} = 64(5y - 3)

at the point

\left( \frac{8}{5}, \frac{6}{5} \right)

and (ii) the y-axis, is equal to ________.

Q92·MathematicsSingle correctJEE Main 2022

Let the tangent to the circle

C_1 : x^2 + y^2 = 2

at the point

M(-1, 1)

intersect the circle

C_2 : (x - 3)^2 + (y - 2)^2 = 5

, at two distinct points A and B. If the tangents to

C_2

at the points A and B intersect at N, then the area of the triangle ANB is equal to :

(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{1}{6}$
(D) $\frac{5}{3}$

Q93·MathematicsSingle correctJEE Main 2022

If the tangents drawn at the point O(0, 0) and

\mathrm{P}\left(1+\sqrt{5},2\right)

on the circle

x^{2}+y^{2}-2x-4y=0

intersect at the point Q, then the area of the triangle OPQ is equal to

(A) $\frac{3+\sqrt{5}}{2}$
(B) $\frac{4+2\sqrt{5}}{2}$
(C) $\frac{5+3\sqrt{5}}{2}$
(D) $\frac{7+3\sqrt{5}}{2}$

Q94·MathematicsNumericalJEE Main 2022

Let the lines

y + 2x = \sqrt{11} + 7\sqrt{7}

and

2y + x = 2\sqrt{11} + 6\sqrt{7}

be normal to a circle

C : (x - h)^2 + (y - k)^2 = r^2.

If the line

\sqrt{11}y - 3x = \frac{5\sqrt{77}}{3} + 11

is tangent to the circle C, then the value of

(5h - 8k)^2 + 5r^2

is equal to ______.

Q95·MathematicsNumericalJEE Main 2022

If one of the diameters of the circle

x^2 + y^2 - 2\sqrt{2}x - 6\sqrt{2}y + 14 = 0

is a chord of the circle

\left(x - 2\sqrt{2}\right)^2 + \left(y - 2\sqrt{2}\right)^2 = r^2

, then the value of

r^2

is equal to

Q96·MathematicsSingle correctJEE Main 2022

The set of values of k for which the circle

C:\;4x^{2}+4y^{2}-12x+8y+k=0

lies inside the fourth quadrant and the point

\left(1,-\dfrac{1}{3}\right)

lies on or inside the circle C is :

(A)An empty set
(B) $\left(6,\dfrac{95}{9}\right]$
(C) $\left[\dfrac{80}{9},10\right)$
(D) $\left(9,\dfrac{92}{9}\right]$

Q97·MathematicsNumericalJEE Main 2022

A rectangle R with end points of the one of its dies as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x

-

y + 4 = 0, then the area of R is _________.

Q98·MathematicsSingle correctJEE Main 2022

Let C be a circle passing through the points A(2, -1) and B(3, 4). The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle

(x - 5)^{2} + (y - 1)^{2} = \frac{13}{2}

, then

r^{2}

is equal to :

(A)32
(B) $\frac{65}{2}$
(C) $\frac{61}{2}$
(D)30

Q99·MathematicsSingle correctJEE Main 2022

Let a circle C touch the lines

L_1 : 4x - 3y + K_1 = 0

and

L_2 : 4x - 3y + K_2 = 0

,

K_1

,

K_2 \in R

. If a line passing through the centre of the circle C intersects

L_1

at

(-1, 2)

and

L_2

at

(3, -6)

, then the equation of the circle C is

(A) $(x - 1)^2 + (y - 2)^2 = 4$
(B) $(x + 1)^2 + (y - 2)^2 = 4$
(C) $(x - 1)^2 + (y + 2)^2 = 16$
(D) $(x - 1)^2 + (y - 2)^2 = 16$

Q100·MathematicsNumericalJEE Main 2022

Let the abscissae of the two points P and Q be the roots of

2x^2 - rx + p = 0

and the ordinates of P and Q be the roots of

x^2 - sx - q = 0

. If the equation of the circle described on PQ as diameter is

2(x^2 + y^2) - 11x - 14y - 22 = 0

, then

2r + s - 2q + p

is equal to

Q101·MathematicsSingle correctJEE Main 2022

A circle touches both the y-axis and the line

x + y = 0

. Then the locus of its center is

(A) $y = \sqrt{2}x$
(B) $x = \sqrt{2}y$
(C) $y^2 - x^2 = 2xy$
(D) $x^2 - y^2 = 2xy$

Q102·MathematicsSingle correctJEE Main 2022

Let

x^2 + y^2 + Ax + By + C = 0

be a circle passing through

(0, 6)

and touching the parabola

y = x^2

at

(2, 4)

. Then

A + C

is equal to_______ .

(A)16
(B)88/5
(C)72
(D) $-8$

Q103·MathematicsNumericalJEE Main 2022

Let a circle

C : (x - h)^{2} + (y - k)^{2} = r^{2}

,

k > 0

, touch the x-axis at

(1, 0)

. If the line

x + y = 0

intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of

h + k + r

is equal to ______.

Q104·MathematicsSingle correctJEE Advanced 2021

Consider a triangle

\Delta

whose two sides lie on the x-axis and the line

x + y + 1 = 0

. If the orthocenter of

\Delta

is

(1, 1)

, then the equation of the circle passing through the vertices of the triangle

\Delta

is

(A) $x^2 + y^2 - 3x + y = 0$
(B) $x^2 + y^2 + x + 3y = 0$
(C) $x^2 + y^2 + 2y - 1 = 0$
(D) $x^2 + y^2 + x + y = 0$

Q105·MathematicsNumericalJEE Advanced 2021

Consider the region

R = \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x \ge 0 \text{ and } y^2 \le 4 - x \right\}

. Let

F

be the family of all circles that are contained in

R

and have centers on the x-axis. Let

C

be the circle that has largest radius among the circles in

F

. Let

(\alpha, \beta)

be a point where the circle

C

meets the curve

y^2 = 4 - x

. The value of α is _____.

Q106·MathematicsNumericalJEE Advanced 2021

Consider the region

R = \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x \ge 0 \text{ and } y^2 \le 4 - x \right\}

. Let

F

be the family of all circles that are contained in

R

and have centers on the x-axis. Let

C

be the circle that has largest radius among the circles in

F

. Let

(\alpha, \beta)

be a point where the circle

C

meets the curve

y^2 = 4 - x

. The radius of the circle C is _____.

Q107·MathematicsNumericalJEE Main 2021

If the variable line

3x + 4y = \alpha

lies between the two circles

(x - 1)^{2} + (y - 1)^{2} = 1

and

(x - 9)^{2} + (y - 1)^{2} = 4

, without intercepting a chord on either circle, then the sum of all the integral values of

\alpha

is ______.

Q108·MathematicsNumericalJEE Main 2021

Let B be the centre of the circle

x^{2} + y^{2} - 2x + 4y + 1 = 0

. Let the tangents at two points P and Q on the circle intersect at the point A(3, 1). Then

8.\left( \frac{\text{area } \Delta APQ}{\text{area } \Delta BPQ} \right)

is equal to ______.

Q109·MathematicsNumericalJEE Main 2021

Let the equation

x^{2} + y^{2} + px + (1 - p)y + 5 = 0

represent circles of varying radius

r \in (0, 5]

. Then the number of elements in the set

S = \{q : q = p^{2} \text{ and } q \text{ is an integer}\}

is _________.

Q110·MathematicsNumericalJEE Main 2021

Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and

C_{1}(\alpha,\ \beta)

and

C_{2}\,(\gamma,\ \delta)

,

C_{1} \neq C_{2}

are their centres, then

|(\alpha + \beta)\,(\gamma + \delta)|

is equal to ____________ .

Q111·MathematicsSingle correctJEE Main 2021

If a line along a chord of the circle

4x^2 + 4y^2 + 120x + 675 = 0

, passes through the point

(-30, 0)

and is tangent to the parabola

y^2 = 30x

, then the length of this chord is :

(A) $5$
(B) $7$
(C) $5\sqrt{3}$
(D) $3\sqrt{5}$

Q112·MathematicsNumericalJEE Main 2021

The locus of a point, which moves such that the sum of squares of its distances from the points

(0, 0)

,

(1, 0)

,

(0, 1)

(1, 1)

is 18 units, is a circle of diameter d. Then

d^{2}

is equal to ________.

Q113·MathematicsSingle correctJEE Main 2021

A circle C touches the line

x = 2y

at the point

(2,1)

and intersects the circle

C_{1} : x^{2} + y^{2} + 2y - 5 = 0

at two points P and Q such that PQ is a diameter of

C_{1}

. Then the diameter of C is :

(A) $7\sqrt{5}$
(B) $15$
(C) $\sqrt{285}$
(D) $4\sqrt{15}$

Q114·MathematicsSingle correctJEE Main 2021

Let

A = \left\{ (x, y) \in R \times R \,\middle|\, 2x^2 + 2y^2 - 2x - 2y = 1 \right\}

,

B = \left\{ (x, y) \in R \times R \,\middle|\, 4x^2 + 4y^2 - 16y + 7 = 0 \right\}

and

C = \left\{ (x, y) \in R \times R \,\middle|\, x^2 + y^2 - 4x - 2y + 5 \le r^2 \right\}

. Then the minimum value of

|r|

such that

A \cup B \subseteq C

is equal to :

(A) $1 + \sqrt{5}$
(B) $\frac{3 + 2\sqrt{5}}{2}$
(C) $\frac{3 + \sqrt{10}}{2}$
(D) $\frac{2 + \sqrt{10}}{2}$

Q115·MathematicsSingle correctJEE Main 2021

Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept 6√5 on the x-axis. Then the radius of the circle C is equal to :

(A)9
(B)√82
(C)8
(D)√53

Q116·MathematicsSingle correctJEE Main 2021

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set

\left\{P, Q\right\}

is equal to :

(A) $\left\{\left(-1,5\right),\left(5,1\right)\right\}$
(B) $\left\{\left(2+2\sqrt{2},3+\sqrt{5}\right),\left(2-2\sqrt{2},3-\sqrt{5}\right)\right\}$
(C) $\left\{\left(2+2\sqrt{2},3-\sqrt{5}\right),\left(2-2\sqrt{2},3+\sqrt{5}\right)\right\}$
(D) $\left\{\left(4,0\right),\left(0,6\right)\right\}$

Q117·MathematicsSingle correctJEE Main 2021

Two tangents are drawn from the point

P\left(-1, 1\right)

to the circle

x^2 + y^2 - 2x - 6y + 6 = 0

. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to :

(A) $3\left(\sqrt{2} - 1\right)$
(B) $\left(3\sqrt{2} + 2\right)$
(C)4
(D)2

Q118·MathematicsSingle correctJEE Main 2021

Let the circle

S : 36x^2 + 36y^2 - 108x + 120y + C = 0

be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines,

x - 2y = 4

and

2x - y = 5

lies inside the circle S, then :

(A)81 < C < 156
(B)100 < C < 165
(C)100 < C < 156
(D) $\frac{25}{9} < C < \frac{13}{3}$

Q119·MathematicsSingle correctJEE Main 2021

Let

r_{1}

and

r_{2}

be the radii of the largest and smallest circles, respectively, which pass through the point

(-4,1)

and having their centres on the circumference of the circle

x^{2}+y^{2}+2x+4y-4=0

. If

\frac{r_{1}}{r_{2}}=a+b\sqrt{2},

then

a+b

is equal to :

(A) $5$
(B) $11$
(C) $7$
(D) $3$

Q120·MathematicsSingle correctJEE Main 2021

Choose the correct statement about two circles whose equations are given below :

x^{2} + y^{2} - 10x - 10y + 41 = 0

x^{2} + y^{2} - 22x - 10y + 137 = 0

(A)circles have same centre
(B)circles have no meeting point
(C)circles have only one meeting point
(D)circles have two meeting points

Q121·MathematicsSingle correctJEE Main 2021

Let

S_{1} : x^{2} + y^{2} = 9

and

S_{2} : (x-2)^{2} + y^{2} = 1

. Then the locus of center of a variable circle S which touches

S_{1}

internally and

S_{2}

externally always passes through the points :

(A) $\left(0, \pm\sqrt{3}\right)$
(B) $\left(\frac{1}{2}, \pm\frac{\sqrt{5}}{2}\right)$
(C) $\left(2, \pm\frac{3}{2}\right)$
(D) $\left(1, \pm 2\right)$

Q122·MathematicsSingle correctJEE Main 2021

For the four circles M, N, O and P, following four equations are given : Circle M :

x^{2} + y^{2} = 1

Circle N :

x^{2} + y^{2} - 2x = 0

Circle O :

x^{2} + y^{2} - 2x - 2y + 1 = 0

Circle P :

x^{2} + y^{2} - 2y = 0

If the centre of circle M is joined with centre of the circle N, further centre of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a :

(A)Rhombus
(B)Square
(C)Rectangle
(D)Parallelogram

Q123·MathematicsSingle correctJEE Main 2021

Choose the incorrect statement about the two circles whose equations are given below :

x^2 + y^2 - 10x - 10y + 41 = 0

and

x^2 + y^2 - 16x - 10y + 80 = 0

(A)Distance between two centres is the average of radii of both the circles.
(B)Both circles' centres lie inside region of one another.
(C)Both circles pass through the centre of each other.
(D)Circles have two intersection points.

Q124·MathematicsSingle correctJEE Main 2021

The line

2x - y + 1 = 0

is a tangent to the circle at the point

(2, 5)

and the centre of the circle lies on

x - 2y = 4

. Then, the radius of the circle is:

(A) $3\sqrt{5}$
(B) $5\sqrt{3}$
(C) $5\sqrt{4}$
(D) $4\sqrt{5}$

Q125·MathematicsSingle correctJEE Main 2021

Two tangents are drawn from a point P to the circle

x^{2}+y^{2}-2x-4y+4=0

, such that the angle between these tangents is

\tan^{-1}\left(\frac{12}{5}\right)

, where

\tan^{-1}\left(\frac{12}{5}\right) \in (0, \pi)

. If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of

\Delta PAB

and

\Delta CAB

is :

(A) $11 : 4$
(B) $9 : 4$
(C) $3 : 1$
(D) $2 : 1$

Q126·MathematicsNumericalJEE Main 2021

The minimum distance between any two points

P_1

and

P_2

while considering point

P_1

on one circle and point

P_2

on the other circle for the given circles' equations

x^2 + y^2 - 10x - 10y + 41 = 0

x^2 + y^2 - 24x - 10y + 160 = 0

is ______ .

Q127·MathematicsSingle correctJEE Main 2021

Let the tangent to the circle

x^{2}+y^{2}=25

at the point

R(3, 4)

meet x-axis and y-axis at point P and Q, respectively. If

r

is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then

r^{2}

is equal to

(A) $\frac{529}{64}$
(B) $\frac{125}{72}$
(C) $\frac{625}{72}$
(D) $\frac{585}{66}$

Q128·MathematicsNumericalJEE Main 2021

Let ABCD be a square of side of unit length. Let a circle C₁ centered at A with unit radius is drawn. Another circle C₂ which touches C₁ and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C₂ meet the side AB at E. If the length of EB is

\alpha+\sqrt{3}\beta

, where α, β are integers, then α + β is equal to________.

Q129·MathematicsSingle correctJEE Main 2021

Let the lengths of intercepts on x-axis and y-axis made by the circle

x^{2}+y^{2}+ax+2ay+c=0

,

(a < 0)

be

2\sqrt{2}

and

2\sqrt{5}

, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line

x + 2y = 0

, is euqal to :

(A) $\sqrt{11}$
(B) $\sqrt{7}$
(C) $\sqrt{6}$
(D) $\sqrt{10}$

Q130·MathematicsSingle correctJEE Main 2021

If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle,

x^{2} + y^{2} = 1

is a circle of the radius r, then r is equal to :

(A) $\frac{1}{4}$
(B) $\frac{1}{2}$
(C)1
(D) $\frac{1}{3}$

Q131·MathematicsSingle correctJEE Main 2021

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ ⊥ OB. Then, the area of the triangle PQB (in square units) is:

(A) $26\sqrt{3}$
(B) $24\sqrt{2}$
(C) $24\sqrt{3}$
(D) $26\sqrt{2}$

Q132·MathematicsSingle correctJEE Main 2021

Let A (1, 4) and B(1, −5) be two points. Let P be a point on the circle

(x - 1)^{2} + (y - 1)^{2} = 1

such that

(PA)^{2} + (PB)^{2}

have maximum value, then the points, P, A and B lie on :

(A)a parabola
(B)a straight line
(C)a hyperbola
(D)an ellipse

Q133·MathematicsNumericalJEE Main 2021

If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle

(x-2)^{2}+(y-3)^{2}=25

at the point (5,7) is A, then 24A is equal to________.

Q134·MathematicsNumericalJEE Main 2021

Let a point P be such that its distance from the point (5,0) is thrice the distance of P from the point

(-5,0)

. If the locus of the point P is a circle of radius r, then

4r^{2}

is equal to

Q135·MathematicsNumericalJEE Main 2021

If one of the diameters of the circle

x^2 + y^2 - 2x - 6y + 6 = 0

is a chord of another circle ‘C’ whose center is at (2,1), then its radius is _______

Q136·MathematicsIntegerJEE Advanced 2020

Let

O

be the centre of the circle

x^{2} + y^{2} = r^{2}

, where

r > \frac{\sqrt{5}}{2}

. Suppose

PQ

is a chord of this circle and the equation of the line passing through

P

and

Q

is

2x + 4y = 5

. If the centre of the circumcircle of the triangle

OPQ

lies on the line

x + 2y = 4

, then the value of

r

is ________

Q137·MathematicsSingle correctJEE Main 2020

The centre of the circle passing through the point

(0, 1)

and touching the parabola

y = x^2

at the point

(2, 4)

is:

(A) $\left(\frac{-53}{10}, \frac{16}{5}\right)$
(B) $\left(\frac{6}{5}, \frac{53}{10}\right)$
(C) $\left(\frac{3}{10}, \frac{16}{5}\right)$
(D) $\left(\frac{-16}{5}, \frac{53}{10}\right)$

Q138·MathematicsSingle correctJEE Main 2020

If the length of the chord of the circle,

x^2+y^2=r^2\ (r>0)

along the line

y-2x=3

is r, the

r^2

is equal to:

(A) $12$
(B) $\dfrac{12}{5}$
(C) $\dfrac{9}{5}$
(D) $\dfrac{24}{5}$

Q139·MathematicsSingle correctJEE Main 2020

The circle passing through the intersection of the circles,

x^{2} + y^{2} - 6x = 0

and

x^{2} + y^{2} - 4y = 0

, having its centre on the line,

2x - 3y + 12 = 0

, also passes through the point:

(A)(−1, 3)
(B)(−3, 6)
(C)(−3, 1)
(D)(1, −3)

Q140·MathematicsNumericalJEE Main 2020

Let PQ be a diameter of the circle

x^{2} + y^{2} = 9

. If α and β are the lengths of the perpendiculars from P and Q on the straight line, x + y = 2 respectively, then the maximum value of αβ is……

Q141·MathematicsNumericalJEE Main 2020

The diameter of the circle, whose centre lies on the lines

x + y = 2

in the first quadrant and which touches both the lines

x = 3

and

y = 2

, is __________.

Q142·MathematicsSingle correctJEE Main 2020

Let the latus rectum of the parabola

y^2 = 4x

be the common chord to the circles

C_1

and

C_2

each of them having radius

2\sqrt{5}

. Then, the distance between the centres of the circles

C_1

and

C_2

is:

(A) $4\sqrt{5}$
(B) $8\sqrt{5}$
(C) $8$
(D) $12$

Q143·MathematicsNumericalJEE Main 2020

If the curves,

x^{2} - 6x + y^{2} + 8 = 0

and

x^{2} - 8y + y^{2} + 16 - k = 0

,

(k > 0)

touch each other at a point, then the largest value of k is ______.

Q144·MathematicsSingle correctJEE Main 2020

A circle touches the y - axis at the point

(0,4)

and passes through the point

(2,0)

. Which of the following lines is not a tangent to this circle?

(A) $3x-4y-24=0$
(B) $3x+4y-6=0$
(C) $4x-3y+17=0$
(D) $4x+3y-8=0$

Q145·MathematicsSingle correctJEE Main 2020

If a line,

y=mx+c

is a tangent to the circle,

(x-3)^{2}+y^{2}=1

and it is perpendicular to a line

L_{1}

, where

L_{1}

is the tangent to the circle,

x^{2}+y^{2}=1

at the point

\left(\dfrac{1}{\sqrt{2}},\dfrac{1}{\sqrt{2}}\right)

; then:

(A) $c^{2}+7c+6=0$
(B) $c^{2}-6c+7=0$
(C) $c^{2}+6c+7=0$
(D) $c^{2}-7c+6=0$

Q146·MathematicsSingle correctJEE Main 2020

Let the tangents drawn from the origin to the circle,

x^{2}+y^{2}-8x-4y+16=0

touch it at the points A and B. Then

(AB)^{2}

is equal to:

(A) $\dfrac{56}{5}$
(B) $\dfrac{32}{5}$
(C) $\dfrac{64}{5}$
(D) $\dfrac{52}{5}$

Q147·MathematicsSingle correctJEE Advanced 2019

Answer the following by appropriately matching the lists based on the information given in the paragraph Let the circles

C_1 : x^{2} + y^{2} = 9

and

C_2 : (x - 3)^{2} + (y - 4)^{2} = 16

, intersect at the points X and Y. Suppose that another circle

C_3 : (x - h)^{2} + (y - k)^{2} = r^{2}

satisfies and following conditions: (i) centre of

C_3

is collinear with the centres of

C_1

and

C_2

(ii)

C_1

and

C_2

both lie inside

C_3

, and (iii)

C_3

touches

C_1

at M and

C_2

at N. Let the line through X and Y intersect

C_3

at Z and W, and let a common tangent of

C_1

and

C_3

be a tangent to the parabola

x^{2} = 8\alpha y

. There are some expressions given in the List–I whose values are given in List–II below: Which of the following is the only CORRECT combination?

List – I	List – II
I. $2h + k$	P.6
II. $\dfrac{\text{Length of ZW}}{\text{Length of XY}}$	Q. $\sqrt{6}$
III. $\dfrac{\text{Area of triangle MZN}}{\text{Area of triaangle ZMW}}$	R. $\dfrac{5}{4}$
IV. $\alpha$	S. $\dfrac{21}{5}$
	T. $2\sqrt{6}$
	U. $\dfrac{10}{3}$

(A)(II), (T)
(B)(II), (Q)
(C)(I), (U)
(D)(I), (S)

Q148·MathematicsSingle correctJEE Advanced 2019

Answer the following by appropriately matching the lists based on the information given in the paragraph Let the circles

C_1 : x^{2} + y^{2} = 9

and

C_2 : (x - 3)^{2} + (y - 4)^{2} = 16

, intersect at the points X and Y. Suppose that another circle

C_3 : (x - h)^{2} + (y - k)^{2} = r^{2}

satisfies and following conditions: (i) centre of

C_3

is collinear with the centres of

C_1

and

C_2

(ii)

C_1

and

C_2

both lie inside

C_3

, and (iii)

C_3

touches

C_1

at M and

C_2

at N. Let the line through X and Y intersect

C_3

at Z and W, and let a common tangent of

C_1

and

C_3

be a tangent to the parabola

x^{2} = 8\alpha y

. There are some expressions given in the List–I whose values are given in List–II below: Which of the following is the only INCORRECT combination?

List – I	List – II
I. $2h + k$	P.6
II. $\dfrac{\text{Length of ZW}}{\text{Length of XY}}$	Q. $\sqrt{6}$
III. $\dfrac{\text{Area of triangle MZN}}{\text{Area of triaangle ZMW}}$	R. $\dfrac{5}{4}$
IV. $\alpha$	S. $\dfrac{21}{5}$
	T. $2\sqrt{6}$
	U. $\dfrac{10}{3}$

(A)(IV), (S)
(B)(III), (R)
(C)(IV), (U)
(D)(I), (P)

Q149·MathematicsSingle correctJEE Advanced 2019

A line

y = mx + 1

intersects the circle

(x - 3)^2 + (y + 2)^2 = 25

at the points P and Q. If the midpoint of the line segment PQ has x-coordinate

-\frac{3}{5}

, then which one of the following options is correct?

(A) $4 \le m < 6$
(B) $-3 \le m < -1$
(C) $2 \le m < 4$
(D) $6 \le m < 8$

Q150·MathematicsNumericalJEE Advanced 2019

Let the point B be the reflection of the point A(2, 3) with respect to line

8x - 6y - 23 = 0

. Let

\Gamma_A

and

\Gamma_B

be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles

\Gamma_A

and

\Gamma_B

such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is ____

Q151·MathematicsSingle correctJEE Main 2019

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point :

(A)(3, 5)
(B)(1, 5)
(C)(3, 10)
(D)(2, 3)

Q152·MathematicsSingle correctJEE Main 2019

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is :

(A) $\dfrac{13}{2}$
(B) $\dfrac{120}{13}$
(C) $\dfrac{13}{5}$
(D) $\dfrac{60}{13}$

Q153·MathematicsSingle correctJEE Main 2019

The locus of the centres of the circles, which touch the circle,

x^{2} + y^{2} = 1

externally, also touch the y-axis and lie in the first quadrant is

(A) $x = \sqrt{1 + 2y}, y \ge 0$
(B) $y = \sqrt{1 + 4x}, x \ge 0$
(C) $x = \sqrt{1 + 4y}, y \ge 0$
(D) $y = \sqrt{1 + 2x}, x \ge 0$

Q154·MathematicsSingle correctJEE Main 2019

If the circles

x^{2}+y^{2}+5Kx+2y+K=0

and

2(x^{2}+y^{2})+2Kx+3y-1=0

, (K

\in

R), intersect at the points P and Q, then the line 4x + 5y − K = 0 passes through P and Q for:

(A)exactly one value of K
(B)not value of K
(C)infinitely many values of K
(D)exactly two values of K

Q155·MathematicsSingle correctJEE Main 2019

The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, −3), then its radius is

(A) $3\sqrt{2}$
(B)3
(C)2
(D) $2\sqrt{2}$

Q156·MathematicsSingle correctJEE Main 2019

If a tangent to the circle

x^{2}+y^{2}=1

intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:

(A) $x^{2}+y^{2}-16x^{2}y^{2}=0$
(B) $x^{2}+y^{2}-2x^{2}y^{2}=0$
(C) $x^{2}+y^{2}-4x^{2}y^{2}=0$
(D) $x^{2}+y^{2}-2xy=0$

Q157·MathematicsSingle correctJEE Main 2019

The area in sq. units) of the smaller of the two circles that touch the parabola,

y^2=4x

at the points

(1,2)

and the axis is

(A) $4\pi\left(2-\sqrt{2}\right)$
(B) $8\pi\left(3-2\sqrt{2}\right)$
(C) $4\pi\left(3+\sqrt{2}\right)$
(D) $8\pi\left(2-\sqrt{2}\right)$

Q158·MathematicsSingle correctJEE Main 2019

The common tangent to the circles

x^{2} + y^{2} = 4

and

x^{2} + y^{2} + 6x + 8y - 24 = 0

also passes through the point

(A) $(-4, 6)$
(B) $(6. -2)$
(C) $(-6, 4)$
(D) $(4, -2)$

Q159·MathematicsSingle correctJEE Main 2019

The sum of the squares of the lengths of the chords intercepted on the circle,

x^{2}+y^{2}=16

, by the lines,

x+y=n, n\in N

, where N is the set of all natural numbers is:

(A)320
(B)160
(C)105
(D)210

Q160·MathematicsSingle correctJEE Main 2019

Then tangent and the normal lines at the point

(\sqrt{3},1)

to the circle

x^{2}+y^{2}=4

and the x-axis form a triangle. The area of this triangle (in square units) is:

(A) $\frac{1}{\sqrt{3}}$
(B) $\frac{4}{\sqrt{3}}$
(C) $\frac{1}{3}$
(D) $\frac{2}{\sqrt{3}}$

Q161·MathematicsSingle correctJEE Main 2019

If a variable line,

3x+4y-\lambda=0

is such that the two circles

x^{2}+y^{2}-2x-2y+1=0

and

x^{2}+y^{2}-18x-2y+78=0

are on its opposite sides, then the set of all values of

\lambda

is the interval:

(A) $(2, 17)$
(B) $[13, 23]$
(C) $[12, 21]$
(D) $(23, 31)$

Q162·MathematicsSingle correctJEE Main 2019

If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is :

(A) $(x^{2} + y^{2})^{2} = 4R^{2}x^{2}y^{2}$
(B) $(x^{2} + y^{2})^{3} = 4R^{2}x^{2}y^{2}$
(C) $(x^{2} + y^{2})^{2} = 4Rx^{2}y^{2}$
(D) $(x^{2} + y^{2})(x + y) = R^{2}xy$

Q163·MathematicsSingle correctJEE Main 2019

Let

C_1

and

C_2

be the centres of the circles

x^{2} + y^{2} - 2x - 2y - 2 = 0

and

x^{2} + y^{2} - 6x - 6y + 14 = 0

respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral

PC_1QC_2

is:

(A)8
(B)6
(C)9
(D)4

Q164·MathematicsSingle correctJEE Main 2019

A circle cuts a chord of length 4a on the x – axis and passes through a point on the y – axis, distant 2b from the origin. Then the locus of the center of this circle, is:

(A)a hyperbola
(B)an ellipse
(C)a straight line
(D)a parabola

Q165·MathematicsSingle correctJEE Main 2019

Two circles with equal radii intersecting at the points (0, 1) and (0, −1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

(A) $1$
(B) $2$
(C) $2\sqrt{2}$
(D) $\sqrt{2}$

Q166·MathematicsSingle correctJEE Main 2019

The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is:

(A) $\frac{\sqrt{5}}{2}$
(B) $2\sqrt{5}$
(C) $\frac{\sqrt{5}}{4}$
(D) $4\sqrt{5}$

Q167·MathematicsSingle correctJEE Main 2019

A square is inscribed in the circle

x^{2}+y^{2}-6x+8y-103=0

with its sides parallel to the coordinate axes. Then the distance of the vertex of the square which is nearest to the origin is:

(A) $6$
(B) $\sqrt{137}$
(C) $\sqrt{41}$
(D) $13$

Q168·MathematicsSingle correctJEE Main 2019

If a circle C passing through the point (4, 0) touches the circle

x^2 + y^2 + 4x - 6y = 12

externally at the point (1, −1), then the radius of C is:

(A) $2\sqrt{5}$
(B)4
(C)5
(D) $\sqrt{57}$

Q169·MathematicsSingle correctJEE Main 2019

Three circles of radii a, b, c ( a < b < c ) touch each other externally. If they have x – axis as a common tangent, then:

(A) $\dfrac{1}{\sqrt{a}}=\dfrac{1}{\sqrt{b}}+\dfrac{1}{\sqrt{c}}$
(B) $\dfrac{1}{\sqrt{b}}=\dfrac{1}{\sqrt{a}}+\dfrac{1}{\sqrt{c}}$
(C)a, b, c are in A.P.
(D) $\sqrt{a},\sqrt{b},\sqrt{c}$ are in A.P.

Q170·MathematicsSingle correctJEE Main 2019

If the circles

x^{2} + y^{2} - 16x - 20y + 164 = r^{2}

and

(x-4)^{2} + (y-7)^{2} = 36

intersect at two distinct points, then:

(A) $0 < r < 1$
(B) $1 < r < 11$
(C) $r > 11$
(D) $r = 11$

Q171·MathematicsSingle correctJEE Advanced 2018

PARAGRAPH "X" Let S be the circle in the x-y plane defined by the equation

x^{2} + y^{2} = 4

. (There are two questions based on PARAGRAPH "X", the question given below is one of them) Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

(A) $(x+y)^{2} = 3xy$
(B) $x^{2/3} + y^{2/3} = 2^{4/3}$
(C) $x^{2} + y^{2} = 2xy$
(D) $x^{2} + y^{2} = x^{2}y^{2}$

Q172·MathematicsMultiple correctJEE Advanced 2018

Let T be the line passing through the points P(

-2

, 7) and Q(2,

-5

). Let

F_{1}

be the set of all pairs of circles (

S_{1}

,

S_{2}

) such that T is tangent to

S_{1}

at P and tangent to

S_{2}

at Q, and also such that

S_{1}

and

S_{2}

touch each other at a point, say, M. Let

E_{1}

be the set representing the locus of M as the pair (

S_{1}

,

S_{2}

) varies in

F_{1}

. Let the set of all straight line segments joining a pair of distinct points of

E_{1}

and passing through the point R(1, 1) be

F_{2}

. Let

E_{2}

be the set of the mid-points of the line segments in the set

F_{2}

. Then, which of the following statement(s) is (are) TRUE ?

(A)The point $(-2, 7)$ lies in $E_{1}$
(B)The point $\left(\frac{4}{5}, \frac{7}{5}\right)$ does NOT lie in $E_{2}$
(C)The point $\left(\frac{1}{2}, 1\right)$ lies in $E_{2}$
(D)The point $\left(0, \frac{3}{2}\right)$ does NOT lie in $E_{1}$

Q173·MathematicsSingle correctJEE Advanced 2018

PARAGRAPH "X" Let S be the circle in the x-y plane defined by the equation

x^{2} + y^{2} = 4

. (There are two questions based on PARAGRAPH "X", the question given below is one of them) Let

E_{1}E_{2}

and

F_{1}F_{2}

be the chords of S passing through the point

P_{0}(1, 1)

and parallel to the x-axis and the y-axis, respectively. Let

G_{1}G_{2}

be the chord of S passing through

P_{0}

and having slope

-1

. Let the tangents to S at

E_{1}

and

E_{2}

meet at

E_{3}

, the tangents to S at

F_{1}

and

F_{2}

meet at

F_{3}

, and the tangents to S at

G_{1}

and

G_{2}

meet at

G_{3}

. Then, the points

E_{3}

,

F_{3}

, and

G_{3}

lie on the curve

(A) $x + y = 4$
(B) $(x-4)^{2} + (y-4)^{2} = 16$
(C) $(x-4)(y-4) = 4$
(D) $xy = 4$

Q174·MathematicsIntegerJEE Advanced 2017

For how many values of

p

, the circle

x^{2} + y^{2} + 2x + 4y - p = 0

and the coordinate axes have exactly three common points ?

Q175·MathematicsSingle correctJEE Advanced 2017

Answer by appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. For

a = \sqrt{2}

, if a tangent is drawn to a suitable conic (Column 1) at the point of contact

(-1, 1)

, then which of the following options is the only CORRECT combination for obtaining its equation ?

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $my = m^{2}x + a$	(P) $\left(\frac{a}{m^{2}}, \frac{2a}{m}\right)$
(II) $x^{2} + a^{2}y^{2} = a^{2}$	(ii) $y = mx + a\sqrt{m^{2} + 1}$	(Q) $\left(\frac{-ma}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}}\right)$
(III) $y^{2} = 4ax$	(iii) $y = mx + \sqrt{a^{2}m^{2} - 1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} + 1}}, \frac{1}{\sqrt{a^{2}m^{2} + 1}}\right)$
(IV) $x^{2} - a^{2}y^{2} = a^{2}$	(iv) $y = mx + \sqrt{a^{2}m^{2} + 1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^{2}m^{2} - 1}}, \frac{-1}{\sqrt{a^{2}m^{2} - 1}}\right)$

(A)(II) (ii) (Q)
(B)(III) (i) (P)
(C)(I) (i) (P)
(D)(I) (ii) (Q)

Q176·MathematicsMultiple correctJEE Advanced 2016

The circle

C_1 : x^{2} + y^{2} = 3

, with centre at O, intersects the parabola

x^{2} = 2y

at the point P in the first quadrant. Let the tangent to the circle

C_1

at P touches other two circles

C_2

and

C_3

at

R_2

and

R_3

, respectively. Suppose

C_2

and

C_3

have equal radii

2\sqrt{3}

and centres

Q_2

and

Q_3

, respectively. If

Q_2

and

Q_3

lie on the y-axis, then

(A) $Q_2Q_3 = 12$
(B) $R_2R_3 = 4\sqrt{6}$
(C)area of the triangle $OR_2R_3$ is $6\sqrt{2}$
(D)area of the triangle $PQ_2Q_3$ is $4\sqrt{2}$

Q177·MathematicsMultiple correctJEE Advanced 2016

Let RS be the diameter of the circle

x^{2} + y^{2} = 1

, where S is the point (1, 0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point(s)

(A) $\left(\frac{1}{3}, \frac{1}{\sqrt{3}}\right)$
(B) $\left(\frac{1}{4}, \frac{1}{2}\right)$
(C) $\left(\frac{1}{3}, -\frac{1}{\sqrt{3}}\right)$
(D) $\left(\frac{1}{4}, -\frac{1}{2}\right)$

Q178·MathematicsMultiple correctJEE Advanced 2014

A circle

S

passes through the point

(0, 1)

and is orthogonal to the circles

(x - 1)^2 + y^2 = 16

and

x^2 + y^2 = 1

. Then

(A)radius of $S$ is $8$
(B)radius of $S$ is $7$
(C)centre of $S$ is $(-7, 1)$
(D)centre of $S$ is $(-8, 1)$

Q179·MathematicsMultiple correctJEE Advanced 2013

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length

2\sqrt{7}

on y-axis is (are)

(A) $x^{2} + y^{2} - 6x + 8y + 9 = 0$
(B) $x^{2} + y^{2} - 6x + 7y + 9 = 0$
(C) $x^{2} + y^{2} - 6x - 8y + 9 = 0$
(D) $x^{2} + y^{2} - 6x - 7y + 9 = 0$

Circles — JEE Previous Year Questions

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