How many questions from Straight Lines appear in JEE?

Straight Lines has appeared in 114 of the last 186 JEE Main and JEE Advanced papers — about 61% of them — contributing 137 questions in total across those papers.

Is Straight Lines an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 61% 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 Straight Lines 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.

Straight Lines JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Main 2026

If a straight line drawn through the point of intersection of the lines

4x + 3y - 1 = 0

and

3x + 4y - 1 = 0

, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is:

(A) $x + y - 7 = 0$
(B) $x + y - 14xy = 0$
(C) $2x + y + 14xy = 0$
(D) $x + 2y - 14xy = 0$

Q2·MathematicsNumericalJEE Main 2026

From the point (−1, −1), two rays are sent making angles of 45° with the line x + y = 0. These rays get reflected from the mirror x + 2y = 1. If the equations of the reflected rays are ax + by = 9 and cx + dy = 7, a, b, c, d ∈ ℤ, then the value of ad + bc is _______.

Q3·MathematicsSingle correctJEE Main 2026

In an equilateral triangle

PQR

, let the vertex

P

be at

(3,5)

and the side

QR

be along the line

x + y = 4

. If the orthocentre of the triangle

PQR

is

(\alpha, \beta)

, then

9(\alpha + \beta)

is equal to:

(A)16
(B)27
(C)36
(D)48

Q4·MathematicsNumericalJEE Main 2026

Let

A, B

be points on the two half-lines

x - \sqrt{3}|y| = \alpha

,

\alpha > 0

at a distance of α from their point of intersection

P

. The line segment

AB

meets the angle bisector of the given half-lines at the point

Q

. If

PQ = \frac{9}{2}

and

R

is the radius of the circumcircle of

\triangle PAB

, then

\frac{\alpha^{2}}{R}

is equal to ______

Q5·MathematicsSingle correctJEE Main 2026

Let the line

L_1 : x + 3 = 0

intersect the lines

L_2 : x - y = 0

and

L_3 : 3x + y = 0

at the points

A

and

B

, respectively. Let the bisector of the obtuse angle between the lines

L_2

and

L_3

intersect the line

L_1

at the point

C

. Then

BC^2 : AC^2

is equal to:

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

Q6·MathematicsSingle correctJEE Main 2026

Let the vertex

A

of a triangle

ABC

be

(1, 2)

, and the mid-point of the side

AB

be

(5, -1)

. If the centroid of this triangle is

(3, 4)

and its circumcenter is

(\alpha, \beta)

, then

21(\alpha + \beta)

is equal to:

(A) $309$
(B) $403$
(C) $497$
(D) $524$

Q7·MathematicsSingle correctJEE Main 2026

Let the mid points of the sides of a triangle ABC be

\left(\frac{5}{2},7\right)

,

\left(\frac{5}{2},3\right)

and

(4,5)

. If its incentre is

(h,k)

, then

3h+k

is equal to :

(A)11
(B)12
(C)13
(D)14

Q8·MathematicsSingle correctJEE Main 2026

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line

x + 2\sqrt{2}\,y = 4

. If the co-ordinates of the vertex A are

(\alpha, \beta)

, then the greatest integer less than or equal to

\left|\alpha + \sqrt{2}\,\beta\right|

is

(A)2
(B)3
(C)5
(D)4

Q9·MathematicsSingle correctJEE Main 2026

Let the angles made with the positive x-axis by two straight lines drawn from the point P(2, 3) and meeting the line

x + y = 6

at a distance

\sqrt{\frac{2}{3}}

from the point P be

\theta_{1}

and

\theta_{2}

. Then the value of

(\theta_{1} + \theta_{2})

is :

(A) $\frac{\pi}{12}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{2}$
(D) $\frac{\pi}{3}$

Q10·MathematicsSingle correctJEE Main 2026

Let A(1, 0), B(2, −1) and

C\left(\frac{7}{3},\frac{4}{3}\right)

be three points. If the equation of the bisector of the angle ABC is

\alpha x+\beta y=5

, then the value of

\alpha^{2}+\beta^{2}

is

(A) $8$
(B) $5$
(C) $13$
(D) $10$

Q11·MathematicsSingle correctJEE Main 2026

Let A(1, 2) and C(−3, −6) be two diagonally opposite vertices of a rhombus, whose sides AD an BC are parallel to the line

7x - y = 14

. If B (

\alpha

,

\beta

) and D(

\gamma

,

\delta

) are the other two vertices, then

|\alpha + \beta + \gamma + \delta|

is equal to :

(A)9
(B)3
(C)6
(D)1

Q12·MathematicsSingle correctJEE Main 2026

A rectangle is formed by the lines x = 0, y = 0, x = 3 and y = 4. Let the line L be perpendicular to 3x + y + 6 = 0 and divide the area of the rectangle into two equal parts. Then the distance of the point

\left(\frac{1}{2}, -5\right)

from the line L is equal to :

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

Q13·MathematicsSingle correctJEE Main 2026

Among the statements (S1) : If A(5, -1) and B(-2, 3) are two vertices of a triangle, whose orthocentre is (0, 0), then its third vertex is (-4, -7) and (S2) : If positive numbers 2a, b, c are three consecutive terms of an A.P., then the lines

ax + by + c = 0

are concurrent at (2, -2),

(A)Only (S1) is correct
(B)Only (S2) is correct
(C)Both are incorrect
(D)Both are correct

Q14·MathematicsSingle correctJEE Main 2026

Let a point A lie between the parallel lines

L_{1}

and

L_{2}

such that its distances from

L_{1}

and

L_{2}

are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC, where the points B and C lie on the lines

L_{1}

and

L_{2}

respectively, is :

(A) $15\sqrt{6}$
(B)27
(C) $21\sqrt{3}$
(D) $12\sqrt{2}$

Q15·MathematicsSingle correctJEE Main 2025

A line passing through the point

P(a,0)

makes an acute angle

\alpha

with the positive x-axis. Let this line be rotated about the point

P

through an angle

\dfrac{\alpha}{2}

in the clock-wise direction. If in the new position, the slope of the line is

2-\sqrt3

and its distance from the origin is

\dfrac{1}{\sqrt2}

, then the value of

3a^2\tan^2\alpha-2\sqrt3

is:

(A) $4$
(B) $6$
(C) $5$
(D) $8$

Q16·MathematicsSingle correctJEE Main 2025

Let

a

be the length of a side of a square

OABC

with

O

being the origin. Its side

OA

makes an acute angle

\alpha

with the positive x-axis and the equations of its diagonals are

(\sqrt3+1)x+(\sqrt3-1)y=0

and

(\sqrt3-1)x-(\sqrt3+1)y+8\sqrt3=0

. Then

a^2

is equal to:

(A) $48$
(B) $32$
(C) $16$
(D) $24$

Q17·MathematicsSingle correctJEE Main 2025

Let ABC be the triangle such that the equations of lines AB and AC be

3y-x=2

and

x+y=2

, respectively, and the points B and C lie on x-axis. If P is the orthocentre of the triangle ABC, then the area of the triangle PBC is equal to

(A)4
(B)10
(C)8
(D)6

Q18·MathematicsSingle correctJEE Main 2025

Consider the lines

x(3\lambda+1)+y(7\lambda+2)=17\lambda+5

,

\lambda

being a parameter, and passing through a point

P

. One of these lines (say

L

) is farthest from the origin. If the distance of

L

from the point

(3,6)

is

d

, then the value of

d^2

is:

(A)20
(B)30
(C)10
(D)15

Q19·MathematicsSingle correctJEE Main 2025

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines

L_1:2x+y+6=0

and

L_2:4x+2y-p=0

, at the points

A

and

B

, respectively. If

AB=\dfrac{9}{\sqrt2}

and the foot of the perpendicular from the point

A

on the line

L_2

is

M

, then

\dfrac{AM}{BM}

is equal to:

(A)5
(B)4
(C)2
(D)3

Q20·MathematicsIntegerJEE Main 2025

Let

A(4,-2)

,

B(1,1)

and

C(9,-3)

be the vertices of a triangle. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is ______.

Q21·MathematicsSingle correctJEE Main 2025

Let the area of the triangle formed by a straight line

L:x+by+c=0

with the co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line

L

makes an angle of

45^\circ

with the positive x-axis, then the value of

b^2+c^2

is:

(A)90
(B)93
(C)97
(D)83

Q22·MathematicsSingle correctJEE Main 2025

Let ABC be a triangle formed by the lines

7x-6y+3=0

,

x+2y-31=0

and

9x-2y-19=0

. Let the point (h,k) be the image of the centroid of

\triangle ABC

in the line

3x+6y-53=0

. Then

h^2+k^2+hk

is equal to

(A)37
(B)47
(C)40
(D)36

Q23·MathematicsSingle correctJEE Main 2025

Let the line

x+y=1

meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is

\dfrac{4}{9}

of the area of the triangle OAB and

AN:NB=\lambda:1

, then the sum of all possible value(s) of

\lambda

is:

(A) $\dfrac{1}{2}$
(B) $\dfrac{13}{6}$
(C) $\dfrac{5}{2}$
(D)2

Q24·MathematicsSingle correctJEE Main 2025

Two equal sides of an isosceles triangle are along

-x+2y=4

and

x+y=4

. If m is the slope of its third side, then the sum, of all possible distinct values of m, is:

(A)-6
(B)12
(C)6
(D) $-2\sqrt{10}$

Q25·MathematicsSingle correctJEE Main 2025

Let

^nC_{r-1}=28

,

^nC_r=56

and

^nC_{r+1}=70

. Let

A(4\cos t, 4\sin t)

,

B(2\sin t, -2\cos t)

and

C(3r-n, r^2-n-1)

be the vertices of a triangle ABC, where t is a parameter. If

(3x-1)^2+(3y)^2=\alpha

, is the locus of the centroid of triangle ABC, then

\alpha

equals:

(A)20
(B)8
(C)6
(D)18

Q26·MathematicsSingle correctJEE Main 2025

Let ABCD be a trapezium whose vertices lie on the parabola

y^2=4x

. Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length

\frac{25}{4}

and it passes through the point

(1, 0)

, then the area of ABCD is:

(A) $\frac{75}{4}$
(B) $\frac{25}{2}$
(C) $\frac{125}{8}$
(D) $\frac{75}{8}$

Q27·MathematicsSingle correctJEE Main 2025

Let the points

\left(\dfrac{11}{2},\alpha\right)

lie on or inside the triangle with sides

x+y=11

,

x+2y=16

and

2x+3y=29

. Then the product of the smallest and the largest values of

\alpha

is equal to:

(A)22
(B)44
(C)33
(D)55

Q28·MathematicsSingle correctJEE Main 2025

Let the lines

3x-4y-\alpha=0

,

8x-11y-33=0

, and

2x-3y+\lambda=0

be concurrent. If the image of the point

(1,2)

in the line

2x-3y+\lambda=0

is

\left(\dfrac{57}{13},\dfrac{-40}{13}\right)

, then

|\alpha\lambda|

is equal to :

(A) $84$
(B) $91$
(C) $113$
(D) $101$

Q29·MathematicsSingle correctJEE Main 2025

A rod of length eight units moves such that its ends

A

and

B

always lie on the lines

x-y+2=0

and

y+2=0

, respectively. If the locus of the point

P

, that divides the rod

AB

internally in the ratio

2:1

is

9(x^2+\alpha y^2+\beta xy+\gamma x+28y)-76=0

, then

\alpha-\beta-\gamma

is equal to :

(A)24
(B)23
(C)21
(D)22

Q30·MathematicsSingle correctJEE Main 2025

Let the area of a

\triangle PQR

with vertices

P(5,4)

,

Q(-2,4)

and

R(a,b)

be 35 square units. If its orthocenter and centroid are

O\left(2,\dfrac{14}{5}\right)

and

C(c,d)

respectively, then

c+2d

is equal to

(A) $\dfrac{7}{3}$
(B) $3$
(C) $2$
(D) $\dfrac{8}{3}$

Q31·MathematicsSingle correctJEE Main 2025

Let triangle PQR be the image of the triangle with vertices

(1,3),(3,1),(2,4)

in the line

x+2y=2

. If the centroid of

\triangle PQR

is

(\alpha,\beta)

, then

15(\alpha-\beta)

equals:

(A)24
(B)19
(C)21
(D)22

Q32·MathematicsIntegerJEE Main 2025

Let

A(6,8)

,

B(10\cos\alpha,-10\sin\alpha)

and

C(-10\sin\alpha,10\cos\alpha)

be the vertices of a triangle. If

L(a,9)

and

G(h,k)

be its orthocenter and centroid respectively, then

(5a-3h+6k+100\sin2\alpha)

is equal to ______.

Q33·MathematicsIntegerJEE Main 2025

Let the distance between two parallel lines be 5 units and a point

P

lie between the lines at a unit distance from one of them. An equilateral triangle

PQR

is formed such that

Q

lies on one of the parallel lines, while

R

lies on the other. Then

(QR)^2

is equal to ______.

Q34·MathematicsSingle correctJEE Main 2024

A variable line L passes through the point

(3, 5)

and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is:

(A)30
(B)25
(C)40
(D)35

Q35·MathematicsSingle correctJEE Main 2024

A ray of light coming from the point

P(1, 2)

gets reflected from the point Q on the x-axis and then passes through the point

R(4, 3)

. If the point

S(h, k)

is such that PQRS is a parallelogram, then

hk^2

is equal to:

(A)80
(B)90
(C)60
(D)70

Q36·MathematicsSingle correctJEE Main 2024

Two vertices of a triangle ABC are

A(3, -1)

and

B(-2, 3)

, and its orthocentre is

P(1, 1)

. If the coordinates of the point C are

(\alpha, \beta)

and the centre of the circle circumscribing the triangle PAB is

(h, k)

, then the value of

(\alpha+\beta)+2(h+k)

equals:

(A) $51$
(B) $81$
(C) $5$
(D) $15$

Q37·MathematicsSingle correctJEE Main 2024

If the line segment joining the points

(5,2)

and

(2,a)

subtends an angle

\dfrac{\pi}{4}

at the origin, then the absolute value of the product of all possible values of

a

is

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

Q38·MathematicsNumericalJEE Main 2024

Let a ray of light passing through the point

(3,10)

reflect on the line

2x+y=6

and the reflected ray pass through the point

(7,2)

. If the equation of the incident ray is

ax+by+1=0

, then

a^2+b^2+3ab

is equal to _______ .

Q39·MathematicsNumericalJEE Main 2024

If the orthocentre of the triangle formed by the lines

2x+3y-1=0

,

x+2y-1=0

and

ax+by-1=0

, is the centroid of another triangle, whose circumcentre and orthocentre respectively are

(3,4)

and

(-6,-8)

, then the value of

|a-b|

is ___

Q40·MathematicsSingle correctJEE Main 2024

The equations of two sides

AB

and

AC

of a triangle

ABC

are

4x+y=14

and

3x-2y=5

, respectively. The point

\left(2,-\dfrac43\right)

divides the third side

BC

internally in the ratio

2:1

. The equation of the side

BC

is:

(A) $x-6y-10=0$
(B) $x-3y-6=0$
(C) $x+3y+2=0$
(D) $x+6y+6=0$

Q41·MathematicsSingle correctJEE Main 2024

If the image of the point

(-4,5)

in the line

x+2y=2

lies on the circle

(x+4)^2+(y-3)^2=r^2

, then

r

is equal to

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

Q42·MathematicsSingle correctJEE Main 2024

If

P(6, 1)

be the orthocentre of the triangle whose vertices are

A(5, -2)

,

B(8, 3)

and

C(h, k)

, then the point C lies on the circle.

(A) $x^2 + y^2 - 65 = 0$
(B) $x^2 + y^2 - 74 = 0$
(C) $x^2 + y^2 - 61 = 0$
(D) $x^2 + y^2 - 52 = 0$

Q43·MathematicsSingle correctJEE Main 2024

Let a variable line of slope

m>0

passing through the point

(4,-9)

intersect the coordinate axes at the points

A

and

B

. The minimum value of the sum of the distances of

A

and

B

from the origin is

(A) $25$
(B) $30$
(C) $15$
(D) $10$

Q44·MathematicsSingle correctJEE Main 2024

Let two straight lines drawn from the origin O intersect the line

3x + 4y = 12

at the points P and Q such that

\Delta OPQ

is an isosceles triangle and

\angle POQ = 90^\circ

. If

l = OP^2 + PQ^2 + QO^2

, then the greatest integer less than or equal to

l

is:

(A) $44$
(B) $48$
(C) $46$
(D) $42$

Q45·MathematicsSingle correctJEE Main 2024

Let

A(-1,1)

and

B(2,3)

be two points and

P

be a variable point above the line

AB

such that the area of

\triangle PAB

is 10. If the locus of

P

is

ax+by=15

, then

5a+2b

is:

(A) $-\dfrac{12}{5}$
(B) $-\dfrac{6}{5}$
(C)4
(D)6

Q46·MathematicsSingle correctJEE Main 2024

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius

r

of the circle passing through the point F and touching the line segments BC and CD satisfies:

(A) $r=1$
(B) $r^2-8r+8=0$
(C) $2r^2-4r+1=0$
(D) $2r^2-8r+7=0$

Q47·MathematicsSingle correctJEE Main 2024

The vertices of a triangle are

A(-1,3)

,

B(-2,2)

and

C(3,-1)

. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

(A) $x-y-(2+\sqrt2)=0$
(B) $-x+y-(2-\sqrt2)=0$
(C) $x+y-(2-\sqrt2)=0$
(D) $x+y+(2-\sqrt2)=0$

Q48·MathematicsNumericalJEE Main 2024

Consider a triangle ABC having the vertices

A(1,2)

,

B(\alpha,\beta)

and

C(\gamma,\delta)

and angles

\angle ABC=\dfrac{\pi}{6}

and

\angle BAC=\dfrac{2\pi}{3}

. If the points B and C lie on the line

y=x+4

, then

\alpha^2+\gamma^2

is equal to

Q49·MathematicsNumericalJEE Main 2024

The lines

L_1,L_2,\ldots,L_{20}

are distinct. For

n=1,2,3,\ldots,10

all the lines

L_{2n-1}

are parallel to each other and all the lines

L_{2n}

pass through a given point P. The maximum number of points of intersection of pairs of lines from the set

\{L_1,L_2,\ldots,L_{20}\}

is equal to __________.

Q50·MathematicsSingle correctJEE Main 2024

Let

\alpha,\beta,\gamma,\delta\in Z

and let

A(\alpha,\beta)

,

B(1,0)

,

C(\gamma,\delta)

and

D(1,2)

be the vertices of a parallelogram ABCD. If

AB=\sqrt{10}

and the points A and C lie on the line

3y=2x+1

, then

2(\alpha+\beta+\gamma+\delta)

is equal to

(A) $10$
(B) $5$
(C) $12$
(D) $8$

Q51·MathematicsSingle correctJEE Main 2024

Let A(

a

,

b

), B(3, 4) and C(

-6

,

-8

) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point P(

2a+3

,

7b+5

) from the line

2x+3y-4=0

measured parallel to the line

x-2y-1=0

is

(A) $\dfrac{15\sqrt{5}}{7}$
(B) $\dfrac{17\sqrt{5}}{6}$
(C) $\dfrac{17\sqrt{5}}{7}$
(D) $\dfrac{4\sqrt{5}}{17}$

Q52·MathematicsNumericalJEE Main 2024

Let

A(-2,-1)

,

B(1,0)

,

C(\alpha,\beta)

and

D(\gamma,\delta)

be the vertices of a parallelogram ABCD. If the point C lies on

2x-y=5

and the point D lies on

3x-2y=6

, then the value of

|\alpha+\beta+\gamma+\delta|

is equal to ______.

Q53·MathematicsSingle correctJEE Main 2024

A line passing through the point

A(9,0)

makes an angle of

30^\circ

with the positive direction of x-axis. If this line is rotated about

A

through an angle of

15^\circ

in the clockwise direction, then its equation in the new position is:

(A) $\dfrac{y}{\sqrt3-2}+x=9$
(B) $\dfrac{x}{\sqrt3-2}+y=9$
(C) $\dfrac{x}{\sqrt3+2}+y=9$
(D) $\dfrac{y}{\sqrt3+2}+x=9$

Q54·MathematicsSingle correctJEE Main 2024

If

x^2-y^2+2hxy+2gx+2fy+c=0

is the locus of a point, which moves such that it is always equidistant from the lines

x+2y+7=0

and

2x-y+8=0

, then the value of

g+c+h-f

equals:

(A) $14$
(B) $6$
(C) $8$
(D) $29$

Q55·MathematicsSingle correctJEE Main 2024

Let

A

be the point of intersection of the lines

3x+2y=14

,

5x-y=6

and

B

be the point of intersection of the lines

4x+3y=8

,

6x+y=5

. The distance of the point

P(5,-2)

from the line

AB

is:

(A) $\dfrac{13}{2}$
(B)8
(C) $\dfrac{5}{2}$
(D)6

Q56·MathematicsSingle correctJEE Main 2024

In a

\triangle ABC

, suppose

y=x

is the equation of the bisector of the angle B and the equation of the side AC is

2x-y=2

. If

2AB=BC

and the point A and B are respectively

(4,6)

and

(\alpha,\beta)

, then

\alpha+2\beta

is equal to

(A) $42$
(B) $39$
(C) $48$
(D) $45$

Q57·MathematicsSingle correctJEE Main 2024

The distance of the point

(2,3)

from the line

2x-3y+28=0

, measured parallel to the line

\sqrt{3}x-y+1=0

, is equal to:

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

Q58·MathematicsSingle correctJEE Main 2024

Let R be the interior region between the lines

3x-y+1=0

and

x+2y-5=0

containing the origin. The set of all values of

a

, for which the points

(a^2,a+1)

lie in R, is :

(A) $(-3,-1)\cup\left(-\dfrac13,1\right)$
(B) $(-3,0)\cup\left(\dfrac13,1\right)$
(C) $(-3,0)\cup\left(\dfrac23,1\right)$
(D) $(-3,-1)\cup\left(\dfrac13,1\right)$

Q59·MathematicsSingle correctJEE Main 2024

The portion of the line

4x+5y=20

in the first quadrant is trisected by the lines

L_1

and

L_2

passing through the origin. The tangent of an angle between the lines

L_1

and

L_2

is:

(A) $\dfrac{8}{5}$
(B) $\dfrac{25}{41}$
(C) $\dfrac{2}{5}$
(D) $\dfrac{30}{41}$

Q60·MathematicsNumericalJEE Main 2024

If the sum of squares of all real values of

\alpha

, for which the lines

2x-y+3=0

,

6x+3y+1=0

and

\alpha x+2y-2=0

do not form a triangle is

p

, then the greatest integer less than or equal to

p

is __________.

Q61·MathematicsNumericalJEE Main 2023

Consider the triangles with vertices

A(2, 1)

,

B(0, 0)

and

C(t, 4)

,

t \in [0, 4]

. If the maximum and the minimum perimeters of such triangles are obtained at

t = \alpha

and

t = \beta

respectively, then

6\alpha + 21\beta

is equal to____

Q62·MathematicsSingle correctJEE Main 2023

If

(\alpha,\beta)

is the orthocentre of the triangle ABC with vertices

A(3,-7)

,

B(-1,2)

and

C(4,5)

, then

9\alpha-6\beta+60

is equal to:

(A)30
(B)25
(C)40
(D)35

Q63·MathematicsSingle correctJEE Main 2023

Let

(\alpha, \beta)

be the centroid of the triangle formed by the lines

15x - y = 82

,

6x - 5y = -4

and

9x + 4y = 17

. Then

\alpha + \beta

and

9\alpha - \beta

are the roots of the equation

(A) $x^2 - 7x + 12 = 0$
(B) $x^2 - 13x + 42 = 0$
(C) $x^2 - 14x + 48 = 0$
(D) $x^2 - 10x + 25 = 0$

Q64·MathematicsSingle correctJEE Main 2023

If the point

\left(\alpha,\dfrac{7\sqrt3}{3}\right)

lies on the curve traced by the mid-points of the line segments of the lines

x\cos\theta+y\sin\theta=7

,

\theta\in\left(0,\dfrac{\pi}{2}\right)

between the co-ordinate axes, then

\alpha

is equal to

(A)7
(B)-7
(C) $-7\sqrt3$
(D) $7\sqrt3$

Q65·MathematicsNumericalJEE Main 2023

If the line

\ell_1:3y-2x=3

is the angular bisector of the lines

\ell_2:x-y+1=0

and

\ell_3:\alpha x+\beta y+17=0

, then

\alpha^2+\beta^2-\alpha-\beta

is equal to

Q66·MathematicsNumericalJEE Main 2023

Let the equations of two adjacent sides of a parallelogram ABCD be

2x-3y=-23

and

5x+4y=23

. If the equation of one diagonal AC is

3x+7y=23

and the distance of A from the other diagonal is

d

, then

50\,d^2

is equal to _______ .

Q67·MathematicsSingle correctJEE Main 2023

Let

C(\alpha,\beta)

be the circumcenter of the triangle formed by the lines

4x+3y=69

,

4y-3x=17

and

x+7y=61

. Then

(\alpha-\beta)^{2}+\alpha+\beta

is equal to

(A)18
(B)17
(C)16
(D)15

Q68·MathematicsSingle correctJEE Main 2023

The straight lines

l_1

and

l_2

pass through the origin and trisect the line segment of the line

L:9x+5y=45

between the axes. If

m_1

and

m_2

are the slopes of the lines

l_1

and

l_2

, then the point of intersection of the line

y=(m_1+m_2)x

with

L

lies on:

(A) $y-x=5$
(B) $y-2x=5$
(C) $6x+y=10$
(D) $2y-x=5$

Q69·MathematicsSingle correctJEE Main 2023

The combined equation of the two lines

ax+by+c=0

and

a'x+b'y+c'=0

can be written as

(ax+by+c)(a'x+b'y+c')=0

. The equation of the angle bisectors of the lines represented by the equation

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

is

(A) $x^2-y^2-10xy=0$
(B) $x^2-y^2+10xy=0$
(C) $3x^2+5xy+2y^2=0$
(D) $3x^2+xy-2y^2=0$

Q70·MathematicsSingle correctJEE Main 2023

If the orthocentre of the triangle, whose vertices are

(1,2)

,

(2,3)

and

(3,1)

is

(\alpha,\beta)

, then the quadratic equation whose roots are

\alpha+4\beta

and

4\alpha+\beta

, is

(A) $x^2-20x+99=0$
(B) $x^2-19x+90=0$
(C) $x^2-22x+120=0$
(D) $x^2-18x+80=0$

Q71·MathematicsSingle correctJEE Main 2023

A straight line cuts off the intercepts

OA=a

and

OB=b

on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of

\dfrac{\pi}{6}

with positive direction of y-axis and the area of

\triangle OAB

is

\dfrac{98}{3}\sqrt3

, then

a^2-b^2

is equal to:

(A) $\dfrac{392}{3}$
(B) $\dfrac{196}{3}$
(C) $98$
(D) $196$

Q72·MathematicsSingle correctJEE Main 2023

A light ray emits from the origin making an angle

30^\circ

with the positive x-axis. After getting reflected by the line

x + y = 1

, if this ray intersects x-axis at

Q

, then the abscissa of

Q

is

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

Q73·MathematicsSingle correctJEE Main 2023

Let

B

and

C

be the two points on the line

y + x = 0

such that

B

and

C

are symmetric with respect to the origin. Suppose

A

is a point on

y - 2x = 2

such that

\triangle ABC

is an equilateral triangle. Then, the area of the

\triangle ABC

is

(A) $\dfrac{10}{\sqrt{3}}$
(B) $3\sqrt{3}$
(C) $2\sqrt{3}$
(D) $\dfrac{8}{\sqrt{3}}$

Q74·MathematicsNumericalJEE Main 2023

A triangle is formed by the X-axis, the Y-axis and the line

3x+4y=60

. Then the number of points

P(a,b)

which lie strictly inside the triangle, where

a

is an integer and

b

is a multiple of

a

, is

Q75·MathematicsSingle correctJEE Main 2023

The equations of two sides of a variable triangle are

x = 0

and

y = 3

, and its third side is a tangent to parabola

y^2 = 6x

. The locus of its circumcentre is:

(A) $4y^2 - 18y - 3x - 18 = 0$
(B) $4y^2 - 18y - 3x + 18 = 0$
(C) $4y^2 - 18y + 3x + 18 = 0$
(D) $4y^2 + 18y + 3x + 18 = 0$

Q76·MathematicsNumericalJEE Main 2023

The equations of the sides AB, BC and CA of a triangle ABC are:

2x+y=0

,

x+py=21a

(a\neq0)

and

x-y=3

respectively. Let

P(2,a)

be the centroid of

\triangle ABC

. Then

(BC)^2

is equal to

Q77·MathematicsSingle correctJEE Main 2023

The equations of the sides AB and AC of a triangle ABC are

(\lambda+1)x+\lambda y=4

and

\lambda x+(1-\lambda)y+\lambda=0

respectively. Its vertex A is on the y-axis and its orthocentre is

(1,2)

. The length of the tangent from the point C to the part of the parabola

y^2=6x

in the first quadrant is:

(A) $4$
(B) $2$
(C) $\sqrt6$
(D) $2\sqrt2$

Q78·MathematicsSingle correctJEE Main 2022

Let the circumcentre of a triangle with vertices

A(a, 3)

,

B(b, 5)

and

C(a, b)

,

ab > 0

be

P(1, 1)

. If the line AP intersects the line BC at the point

Q(k_{1}, k_{2})

, then

k_{1} + k_{2}

is equal to :

(A)2
(B) $\frac{4}{7}$
(C) $\frac{2}{7}$
(D)4

Q79·MathematicsSingle correctJEE Main 2022

Let

A(\alpha, -2)

,

B(\alpha, 6)

and

C\left( \frac{\alpha}{4}, -2 \right)

be vertices of a

\triangle ABC

. If

\left( 5, \frac{\alpha}{4} \right)

is the circumcentre of

\triangle ABC

, then which of the following is NOT correct about

\triangle ABC

:

(A)ares is 24
(B)perimeter is 25
(C)circumradius is 5
(D)inradius is 2

Q80·MathematicsSingle correctJEE Main 2022

Let

m_1

,

m_2

be the slopes of two adjacent sides of a square of side

a

such that

a^2 + 11a + 3\left( m_2^2 + m_2^2 \right) = 220

. If one vertex of the square is

(10(\cos\alpha - \sin\alpha),\ 10(\sin\alpha + \cos\alpha))

, where

\alpha \in \left( 0, \frac{\pi}{2} \right)

and the equation of one diagonal is

(\cos\alpha - \sin\alpha)x + (\sin\alpha + \cos\alpha)y = 10

, then

72(\sin^4\alpha + \cos^4\alpha) + a^2 - 3a + 13

is equal to:

(A) $119$
(B) $128$
(C) $145$
(D) $155$

Q81·MathematicsSingle correctJEE Main 2022

For

t \in (0, 2\pi)

, if ABC is an equilateral triangle with vertices A(sint, −cost), B(cost, sint) and C(a, b) such that its orthocentre lies on a circle with centre

\left(1, \frac{1}{3}\right)

, then

(a^{2} - b^{2})

is equal to :

(A) $\frac{8}{3}$
(B)8
(C) $\frac{77}{9}$
(D) $\frac{80}{9}$

Q82·MathematicsSingle correctJEE Main 2022

Let A(1, 1), B(-4, 3) C(-2, -5) be vertices of a triangle ABC, P be a point on side BC, and

\Delta_{1}

and

\Delta_{2}

be the areas of triangle APB and ABC. Respectively. If

\Delta_{1}:\Delta_{2}=4:7

, then the area enclosed by the lines AP, AC and the x-axis is

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

Q83·MathematicsSingle correctJEE Main 2022

The equations of the sides AB, BC and CA of a triangle ABC are

2x + y = 0

,

x + py = 39

and

x - y = 3

respectively and

P(2, 3)

is its circumcentre. Then which of the following is NOT true :

(A) $(AC)^{2} = 9p$
(B) $(AC)^{2} + p^{2} = 136$
(C) $32 < \text{area } (\Delta ABC) < 36$
(D) $34 < \text{area } (\Delta ABC) < 38$

Q84·MathematicsNumericalJEE Main 2022

The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x − y = 3 respectively. If its orthocentre is (2, a),

-\frac{1}{2}

< a < 2, then p is equal to

Q85·MathematicsSingle correctJEE Main 2022

A line, with the slope greater than one, passes through the point A(4, 3) and intersects the line

x - y - 2 = 0

at the point B. If the length of the line segment AB is

\frac{\sqrt{29}}{3}

, then B also lies on the line :

(A) $2x + y = 9$
(B) $3x - 2y = 7$
(C) $x + 2y = 6$
(D) $2x - 3y = 3$

Q86·MathematicsSingle correctJEE Main 2022

The distance of the origin from the centroid of the triangle whose two sides have the equations

x - 2y + 1 = 0

and

2x - y - 1 = 0

and whose orthocenter is

\left(\frac{7}{3}, \frac{7}{3}\right)

is:

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

Q87·MathematicsSingle correctJEE Main 2022

The distance between the two points A and

A'

which lie on

y = 2

such that both the line segments AB and

A'B

(where B is the point (2, 3)) subtend angle

\frac{\pi}{4}

at the origin, is equal to :

(A)10
(B) $\frac{48}{5}$
(C) $\frac{52}{5}$
(D)3

Q88·MathematicsNumericalJEE Main 2022

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of

7\alpha + 3\beta

is equal to ______.

Q89·MathematicsSingle correctJEE Main 2022

Let a triangle be bounded by the lines

L_1 : 2x + 5y = 10

;

L_2 : -4x + 3y = 12

and the line

L_3

, which passes through the point

P(2, 3)

, intersect

L_2

at A and

L_1

at B. If the point P divides the line-segment AB, internally in the ratio

1 : 3

, then the area of the triangle is equal to

(A) $\frac{110}{13}$
(B) $\frac{132}{13}$
(C) $\frac{142}{13}$
(D) $\frac{151}{13}$

Q90·MathematicsSingle correctJEE Main 2022

In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is

2x + y = 4

. Let the point B lie on the line

x + 3y = 7

. If

(\alpha, \beta)

is the centroid

\Delta ABC

, then

15\left(\alpha + \beta\right)

is equal to :

(A)39
(B)41
(C)51
(D)63

Q91·MathematicsSingle correctJEE Main 2022

Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of

\Delta

PQR is :

(A) $\frac{25}{4\sqrt{3}}$
(B) $\frac{25\sqrt{3}}{2}$
(C) $\frac{25}{\sqrt{3}}$
(D) $\frac{25}{2\sqrt{3}}$

Q92·MathematicsNumericalJEE Main 2022

Let

A\left(\dfrac{3}{\sqrt{a}},\sqrt{a}\right)

a > 0

, be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If D(3 cos

\theta

, a sin

\theta

) is a point in the fourth quadrant such that the maximum area of

\triangle ACD

is 12 square units, then a is equal to ________ .

Q93·MathematicsSingle correctJEE Main 2022

Let the area of the triangle with vertices

A(1, \alpha)

,

B(\alpha, 0)

and

C(0, \alpha)

be 4 sq. units. If the point

(\alpha, -\alpha)

,

(-\alpha, \alpha)

and

(\alpha^{2}, \beta)

are collinear, then

\beta

is equal to

(A)64
(B)-8
(C)-64
(D)512

Q94·MathematicsNumericalJEE Main 2021

Let the points of intersections of the lines x − y + 1 = 0, x − 2y + 3 = 0 and 2x − 5y + 11 = 0 are the mid points of the sides of a triangle ABC. Then the area of the triangle ABC is ________ .

Q95·MathematicsNumericalJEE Main 2021

A man starts walking from the point P(−3,4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then

50\left((PR)^{2} + (RQ)^{2}\right)

is equal to ______ .

Q96·MathematicsSingle correctJEE Main 2021

Let A be the set of all points

(\alpha, \beta)

such that the area of triangle formed by the points

(5, 6)

,

(3, 2)

and

(\alpha, \beta)

is 12 square units. Then the least possible length of a line segment joining the origin to a point in A, is :

(A) $\frac{4}{\sqrt{5}}$
(B) $\frac{16}{\sqrt{5}}$
(C) $\frac{8}{\sqrt{5}}$
(D) $\frac{12}{\sqrt{5}}$

Q97·MathematicsSingle correctJEE Main 2021

If p and q are the lengths of the perpendiculars from the origin on the lines,

x \operatorname{cosec} \alpha - y \sec \alpha = k \cot 2\alpha

and

x \sin \alpha + y \cos \alpha = k \sin 2\alpha

respectively, then

k^{2}

is equal to :

(A) $4p^{2} + q^{2}$
(B) $2p^{2} + q^{2}$
(C) $p^{2} + 2q^{2}$
(D) $p^{2} + 4q^{2}$

Q98·MathematicsSingle correctJEE Main 2021

Let A be a fixed point

(0, 6)

and B be a moving point

(2t, 0)

. Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

(A) $3x^2 - 2y - 6 = 0$
(B) $3x^2 + 2y - 6 = 0$
(C) $2x^2 + 3y - 9 = 0$
(D) $2x^2 - 3y + 9 = 0$

Q99·MathematicsSingle correctJEE Main 2021

Let A(a, 0), B(b, 2b +1) and C(0, b), b ≠ 0, |b| ≠ 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is :

(A) $\frac{-2b}{b+1}$
(B) $\frac{2b}{b+1}$
(C) $\frac{2b^{2}}{b+1}$
(D) $\frac{-2b^{2}}{b+1}$

Q100·MathematicsSingle correctJEE Main 2021

A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that

PC = \sqrt{5}

inches and

\angle PCB = \tan^{-1}(2)

. The acute angle through which the pencil must be rotated about C so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is :

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

Q101·MathematicsSingle correctJEE Main 2021

Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point :

(A) $(2,1)$
(B) $(2,2)$
(C) $(1,3)$
(D) $(1,2)$

Q102·MathematicsSingle correctJEE Main 2021

The point P(a, b) undergoes the following three transformations successively : (a) reflection about the line y = x (b) translation through 2 units along the positive direction of x-axis. (c) rotation through angle

\frac{\pi}{4}

about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point P are

\left( -\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}} \right)

, then the value of 2a + b is equal to :

(A)5
(B)7
(C)13
(D)9

Q103·MathematicsSingle correctJEE Main 2021

Let the equation of the pair of lines, y = px and y = qx, can be written as

\left( y - px \right)\left( y - qx \right) = 0

. Then the equation of the pair of the angle bisectors of the lines

x^2 - 4xy - 5y^2 = 0

is :

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

Q104·MathematicsNumericalJEE Main 2021

Consider a triangle having vertices

A(-2,3)

,

B(1,9)

and

C(3,8)

. If a line L passing through the circum-center of triangle ABC, bisects line BC, and intersects y-axis at point

\left(0,\frac{\alpha}{2}\right)

, then the value of real number

\alpha

is…………

Q105·MathematicsSingle correctJEE Main 2021

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to :

(A) $\frac{9}{\sqrt{2}}$
(B) $7\sqrt{2}$
(C) $2\sqrt{2}$
(D) $3\sqrt{2}$

Q106·MathematicsSingle correctJEE Main 2021

The equation of one of the straight lines which passes through the point

(1,3)

and makes an angles

\tan^{-1}\left(\sqrt{2}\right)

with the straight line,

y + 1 = 3\sqrt{2}\,x

is

(A) $4\sqrt{2}x + 5y - \left(15 + 4\sqrt{2}\right) = 0$
(B) $5\sqrt{2}x + 4y - \left(15 + 4\sqrt{2}\right) = 0$
(C) $4\sqrt{2}x + 5y - 4\sqrt{2} = 0$
(D) $4\sqrt{2}x - 5y - \left(5 + 4\sqrt{2}\right) = 0$

Q107·MathematicsSingle correctJEE Main 2021

The number of integral values of m so that the abscissa of point of intersection of lines

3x + 4y = 9

and

y = mx + 1

is also an integer, is :

(A) $1$
(B) $2$
(C) $3$
(D) $0$

Q108·MathematicsSingle correctJEE Main 2021

Let A(-1, 1), B(3, 4) and C(2, 0) be given three points. A line

y = mx

,

m > 0

, intersects lines AC and BC at point P and Q respectively. Let

A_{1}

and

A_{2}

be the areas of

\Delta ABC

and

\Delta PQC

respectively, such that

A_{1}=3A_{2}

, then the value of m is equal to :

(A) $\frac{4}{15}$
(B) $1$
(C) $2$
(D) $3$

Q109·MathematicsSingle correctJEE Main 2021

The intersection of three lines x−y = 0, x + 2y = 3 and 2x + y = 6 is a:

(A)Equilateral triangle
(B)None of the above
(C)Isosceles triangle
(D)Right angled triangle

Q110·MathematicsSingle correctJEE Main 2021

The image of the point (3,5) in the line

x - y + 1 = 0

, lies on :

(A) $(x - 2)^{2} + (y - 4)^{2} = 4$
(B) $(x - 4)^{2} + (y + 2)^{2} = 16$
(C) $(x - 4)^{2} + (y - 4)^{2} = 8$
(D) $(x - 2)^{2} + (y - 2)^{2} = 12$

Q111·MathematicsSingle correctJEE Main 2021

If the curve x

^{2}

+ 2y

^{2}

= 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is:

(A)2π+tan $^{–1}$ (14 )
(B)2π– tan $^{–1}$ (14 )
(C)2π+tan $^{–1}$ (13 )
(D)2π– tan $^{–1}$ (13 )

Q112·MathematicsSingle correctJEE Main 2021

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is

\frac{1}{4}

. Three stones A, B and C are placed at the points

(1, 1)

,

(2, 2)

and

(4, 4)

respectively. Then which of these stones is/are on the path of the man?

(A)B only
(B)A only
(C)All the three
(D)C only

Q113·MathematicsSingle correctJEE Main 2020

A ray of light coming from the point

(2, 2\sqrt{3})

is incident at an angle

30^\circ

on the line

x=1

at the point A. The ray gets reflected on the line

x=1

and meets

x

-axis at the point B. Then, the line AB passes through the point:

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

Q114·MathematicsSingle correctJEE Main 2020

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point

(-1, -4)

in this line is:

(A) $\left(\frac{11}{5}, \frac{28}{5}\right)$
(B) $\left(\frac{29}{5}, \frac{8}{5}\right)$
(C) $\left(\frac{8}{5}, \frac{29}{5}\right)$
(D) $\left(\frac{29}{5}, \frac{11}{5}\right)$

Q115·MathematicsNumericalJEE Main 2020

If the line,

2x-y+3=0

is at a distance

\dfrac{1}{\sqrt{5}}

and

\dfrac{2}{\sqrt{5}}

from the lines

4x-2y+\alpha=0

and

6x-3y+\beta=0

, respectively, then the sum of all possible values of

\alpha

and

\beta

is ____.

Q116·MathematicsSingle correctJEE Main 2020

If the perpendicular bisector of the line segment joining the points P(1, 4) and Q(k, 3) has y-intercept equal to –4, then a value of k is:

(A) $\sqrt{14}$
(B) $\sqrt{15}$
(C)– 4
(D)– 2

Q117·MathematicsSingle correctJEE Main 2020

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If

\angle BAC = 90°

, and

\mathrm{ar}\left( \Delta ABC \right) = 5\sqrt{5}

sq. units, then the abscissa of the vertex C is:

(A) $1 + 2\sqrt{5}$
(B) $1 + \sqrt{5}$
(C) $2\sqrt{5} - 1$
(D) $2 + \sqrt{5}$

Q118·MathematicsSingle correctJEE Main 2020

If a

\Delta ABC

have vertices A

(-1, 7)

, B

(-7, 1)

and C

(5, -5)

, then its orthocenter has coordinates:

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

Q119·MathematicsSingle correctJEE Main 2020

Let C be the centroid of the triangle with vertices

(3,-1)

,

(1,3)

AND

(2,4)

. Let P be the point of intersection of the lines

x+3y-1=0

and

3x-y+1=0

. Then the line passing through the points C and P also passes through the point:

(A) $(-9,-7)$
(B) $(-9,-6)$
(C) $(7,6)$
(D) $(9,7)$

Q120·MathematicsSingle correctJEE Main 2020

Let two points be A(1, -1) and B(0, 2). If a point

P(x', y')

be such that the area of

\triangle PAB = 5

sq. units and it lies on the line,

3x+y-4\lambda=0

, then a value of

\lambda

is:

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

Q121·MathematicsNumericalJEE Main 2020

Let

A(1,0), B(6,2)

and

C\left(\frac{3}{2}, 6\right)

be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point

\left(-\frac{7}{6}, -\frac{1}{3}\right)

is _________

Q122·MathematicsSingle correctJEE Main 2020

The locus of the mid-points of the perpendiculars drawn from points on the line,

x=2y

to the line

x=y

is:

(A) $7x-5y=0$
(B) $3x-2y=0$
(C) $2x-3y=0$
(D) $5x-7y=0$

Q123·MathematicsSingle correctJEE Main 2019

A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (−1, 1) and (2,3). Then the centroid of this triangle is :

(A) $\left(1,\dfrac{7}{3}\right)$
(B) $\left(\dfrac{1}{3},1\right)$
(C) $\left(\dfrac{1}{3},2\right)$
(D) $\left(\dfrac{1}{3},\dfrac{5}{3}\right)$

Q124·MathematicsSingle correctJEE Main 2019

The equation

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

represents a straight line lying in :

(A)first, third and fourth quadrants
(B)first, second and fourth quadrants
(C)third and fourth quadrants only
(D)second and third quadrants only

Q125·MathematicsSingle correctJEE Main 2019

Lines are drawn parallel to the line 4x − 3y + 2 = 0 at a distance

\dfrac{3}{5}

from the origin. Then which one of the following points lies on any of these lines?

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

Q126·MathematicsSingle correctJEE Main 2019

The region represented by

|x-y|\le 2

and

|x+y|\le 2

is bounded by a:

(A)rhombus of area $8\sqrt{2}$ sq. units
(B)square of area 16 sq. units
(C)rhombus of side length 2 units
(D)square of side length $2\sqrt{2}$ units

Q127·MathematicsSingle correctJEE Main 2019

If the two lines

x + (a-1)y = 1

and

2x + a^{2}y = 1\ (a \in R - \{0,1\})

are perpendicular, then the distance of their point of intersection from the origin is:

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

Q128·MathematicsSingle correctJEE Main 2019

A rectangle is inscribed in a circle with a diameter lying along the line

3y=x+7

. If the two adjacent vertices of the rectangle are

\left(-8,5\right)

and

\left(6,5\right)

then the area of the rectangle (in sq. units) is

(A)72
(B)84
(C)98
(D)56

Q129·MathematicsSingle correctJEE Main 2019

A point on the straight line,

3x+5y=15

which is equidistant from the coordinate, axes will lie only in:

(A)4th quadrant
(B)1st, 2nd and 4th quadrants
(C)1st quadrant
(D)1st and 2nd quadrants

Q130·MathematicsSingle correctJEE Main 2019

Suppose that the points

(h,k)

,

(1,2)

and

(-3,4)

lie on the line

L_{1}

. If a line

L_{2}

passing through the points

(h,k)

and

(4,3)

is perpendicular to

L_{1}

, then

\frac{k}{h}

equals:

(A) $-\frac{1}{7}$
(B) $\frac{1}{3}$
(C)3
(D)0

Q131·MathematicsSingle correctJEE Main 2019

If the straight line,

2x - 3y + 17 = 0

is perpendicular to the line passing through the points (7, 17) and

(15, \beta)

, then

\beta

equals:

(A) $\dfrac{35}{3}$
(B)-5
(C) $-\dfrac{35}{3}$
(D)5

Q132·MathematicsSingle correctJEE Main 2019

If a straight line passing through the point P(−3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is :

(A) $3x - 4y + 25 = 0$
(B) $4x - 3y + 24 = 0$
(C) $x - y + 7 = 0$
(D) $4x + 3y = 0$

Q133·MathematicsSingle correctJEE Main 2019

If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is:

(A) $5x-3y+1=0$
(B) $5x+3y-11=0$
(C) $3x-5y+7=0$
(D) $3x+5y-13=0$

Q134·MathematicsSingle correctJEE Main 2019

A point P moves on the line

2x - 3y + 4 = 0

. If Q(1, 4) and R(3,

-2

) are fixed points, then the locus of the centroid of

\triangle

PQR is a line:

(A)with slope $\dfrac{3}{2}$
(B)parallel to x-axis
(C)with slope $\dfrac{2}{3}$
(D)parallel to y-axis

Q135·MathematicsSingle correctJEE Main 2019

Let the equation of two sides of a triangle be

3x - 2y + 6 = 0

and

4x + 5y - 20 = 0

. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is:

(A) $122y - 26x - 1675 = 0$
(B) $26x + 61y + 1675 = 0$
(C) $122y + 26x + 1675 = 0$
(D) $26x - 122y - 1675 = 0$

Q136·MathematicsSingle correctJEE Main 2019

Consider the set of all lines

px + qy + r = 0

such that

3p + 2q + 4r = 0

. Which one of the following statements is true?

(A)The lines are concurrent at the point $\left(\dfrac{3}{4}, \dfrac{1}{2}\right)$ .
(B)Each line passes through the origin.
(C)The lines are all parallel
(D)The lines are not concurrent

Q137·MathematicsSingle correctJEE Advanced 2013

For

a>b>c>0

, the distance between

(1,1)

and the point of intersection of the lines

ax+by+c=0

and

bx+ay+c=0

is less than

2\sqrt{2}

, then

(A) $a+b-c>0$
(B) $a-b+c<0$
(C) $a-b+c>0$
(D) $a+b-c<0$

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