Straight Lines and Pair of Straight Lines
179 Questions
Start JEE Mains Test
2021
Q101
JEE Mains
Numerical
14 Mar 2026
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 $\Delta $ABC. Then, the area of the $\Delta $ABC is _____________.
Correct Answer: 6
Explanation:
Intersection point of given lines are (1, 2), (7, 5), (2, 3)
$\Delta = {1 \over 2}\left| {\matrix{ 1 & 2 & 1 \cr 7 & 5 & 1 \cr 2 & 3 & 1 \cr } } \right|$
$ = {1 \over 2}[1(5 - 3) - 2(7 - 2) + 1(21 - 10)]$
$ = {1 \over 2}[2 - 10 + 11]$
$\Delta$DEF $ = {1 \over 2}(3) = {3 \over 2}$
$\Delta$ABC = 4$\Delta$DEF $ = 4\left( {{3 \over 2}} \right) = 6$
$\Delta = {1 \over 2}\left| {\matrix{ 1 & 2 & 1 \cr 7 & 5 & 1 \cr 2 & 3 & 1 \cr } } \right|$
$ = {1 \over 2}[1(5 - 3) - 2(7 - 2) + 1(21 - 10)]$
$ = {1 \over 2}[2 - 10 + 11]$
$\Delta$DEF $ = {1 \over 2}(3) = {3 \over 2}$
$\Delta$ABC = 4$\Delta$DEF $ = 4\left( {{3 \over 2}} \right) = 6$
2021
Q102
JEE Mains
Numerical
14 Mar 2026
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 ____________.
Correct Answer: 1250
Explanation:
For minimum $(P R+R Q)$
$R$ lies on $P Q^{\prime}$ (where $Q^{\prime}$ is image of $Q$ in $X$-axis)
$\Rightarrow$ Equation on $P Q^{\prime}$ is
$ 2 x+y+2=0 \Rightarrow R(-1,0) $
$ \therefore $ 50(PR2 + RQ2)
= 50(20 + 5)
= 50(25)
= 1250
2021
Q103
JEE Mains
Numerical
14 Mar 2026
Consider a triangle having vertices A($-$2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point $\left( {0,{\alpha \over 2}} \right)$, then the value of real number $\alpha$ is ________________.
Correct Answer: 9
Explanation:
${\left( {\sqrt {50} } \right)^2} = {\left( {\sqrt {45} } \right)^2} + {\left( {\sqrt 5 } \right)^2}$
$\angle B = 90^\circ $
Circum-center $ = \left( {{1 \over 2},{{11} \over 2}} \right)$
Mid point of BC $ = \left( {2,{{17} \over 2}} \right)$
Line : $\left( {y - {{11} \over 2}} \right) = 2\left( {x - {1 \over 2}} \right) \Rightarrow y = 2x + {9 \over 2}$
Passing through $\left( {0,{\alpha \over 2}} \right)$
${\alpha \over 2} = {9 \over 2} \Rightarrow \alpha = 9$
2021
Q104
JEE Mains
Numerical
14 Mar 2026
A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is _________.
Correct Answer: 80
Explanation:
x2y2 = 1
$ \Rightarrow $ y2 = ${1 \over {{x^2}}}$
$ \Rightarrow $ y = $ \pm {1 \over x}$
Graph of this equation,
$OA \bot OB$
$ \Rightarrow \left( {{1 \over {{p^2}}}} \right)\left( { - {1 \over {{q^2}}}} \right) = - 1$
$ \Rightarrow {p^2}{q^2} = 1$
$P\left( {{{p + q} \over 2},{{{1 \over p} - {1 \over q}} \over 2}} \right)$ midpoint of AB lies
On ${x^2}{y^2} = 1$
$ \Rightarrow {(p + q)^2}{\left( {{1 \over p} - {1 \over q}} \right)^2} = 16$
$ \Rightarrow {(p + q)^2}{(p - q)^2} = 16$
$ \Rightarrow {({p^2} - {q^2})^2} = 16$
$ \Rightarrow {P^2} - {1 \over {{P^2}}} = \pm 4$
$ \Rightarrow {p^4} \pm 4{p^2} - 1 = 0$
$ \Rightarrow {p^2} = {{ \pm 4 \pm \sqrt {20} } \over 2} = \pm 2 \pm \sqrt 5 $
$ \Rightarrow {p^2} = 2 + \sqrt 5 $ or $ - 2 + \sqrt 5 $
$O{B^2} = {p^2} + {1 \over {{p^2}}} = 2 + \sqrt 5 + {1 \over {2 + \sqrt 5 }}$ or $ - 2 + \sqrt 5 + {1 \over { - 2 + \sqrt 5 }} = 2\sqrt 5 $
Area $ = 4\left( {{1 \over 2}} \right)(OA)(OB) = 2{(OB)^2} = 4\sqrt 5 $
$ \Rightarrow $ y2 = ${1 \over {{x^2}}}$
$ \Rightarrow $ y = $ \pm {1 \over x}$
Graph of this equation,
$OA \bot OB$
$ \Rightarrow \left( {{1 \over {{p^2}}}} \right)\left( { - {1 \over {{q^2}}}} \right) = - 1$
$ \Rightarrow {p^2}{q^2} = 1$
$P\left( {{{p + q} \over 2},{{{1 \over p} - {1 \over q}} \over 2}} \right)$ midpoint of AB lies
On ${x^2}{y^2} = 1$
$ \Rightarrow {(p + q)^2}{\left( {{1 \over p} - {1 \over q}} \right)^2} = 16$
$ \Rightarrow {(p + q)^2}{(p - q)^2} = 16$
$ \Rightarrow {({p^2} - {q^2})^2} = 16$
$ \Rightarrow {P^2} - {1 \over {{P^2}}} = \pm 4$
$ \Rightarrow {p^4} \pm 4{p^2} - 1 = 0$
$ \Rightarrow {p^2} = {{ \pm 4 \pm \sqrt {20} } \over 2} = \pm 2 \pm \sqrt 5 $
$ \Rightarrow {p^2} = 2 + \sqrt 5 $ or $ - 2 + \sqrt 5 $
$O{B^2} = {p^2} + {1 \over {{p^2}}} = 2 + \sqrt 5 + {1 \over {2 + \sqrt 5 }}$ or $ - 2 + \sqrt 5 + {1 \over { - 2 + \sqrt 5 }} = 2\sqrt 5 $
Area $ = 4\left( {{1 \over 2}} \right)(OA)(OB) = 2{(OB)^2} = 4\sqrt 5 $
2021
Q105
JEE Mains
Numerical
14 Mar 2026
Let tan$\alpha$, tan$\beta$ and tan$\gamma$; $\alpha$, $\beta$, $\gamma$ $\ne$ ${{(2n - 1)\pi } \over 2}$, n$\in$N be the slopes of three line segments OA, OB and OC, respectively, where O is origin. If circumcentre of $\Delta$ABC coincides with origin and its orthocentre lies on y-axis, then the value of ${\left( {{{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}} \right)^2}$ is equal to ____________.
Correct Answer: 144
Explanation:
Since orthocentre and circumcentre both lies on y-axis.
$ \Rightarrow $ Centroid also lies on y-axis.
$ \Rightarrow $ $\sum {\cos \alpha = 0} $
cos$\alpha$ + cos$\beta$ + cos$\gamma$ = 0
$ \Rightarrow $ cos3 $\alpha$ + cos3 $\beta$ + cos3 $\gamma$ = 3cos$\alpha$cos$\beta$cos$\gamma$
$ \therefore $ ${{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}$
$ = {{4({{\cos }^3}\alpha + {{\cos }^3}\beta + {{\cos }^3}\gamma ) - 3(\cos \alpha + \cos \beta + \cos \gamma )} \over {\cos \alpha \cos \beta \cos \gamma }} = 12$
then, ${\left( {{{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}} \right)^2} = 144$
$ \Rightarrow $ Centroid also lies on y-axis.
$ \Rightarrow $ $\sum {\cos \alpha = 0} $
cos$\alpha$ + cos$\beta$ + cos$\gamma$ = 0
$ \Rightarrow $ cos3 $\alpha$ + cos3 $\beta$ + cos3 $\gamma$ = 3cos$\alpha$cos$\beta$cos$\gamma$
$ \therefore $ ${{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}$
$ = {{4({{\cos }^3}\alpha + {{\cos }^3}\beta + {{\cos }^3}\gamma ) - 3(\cos \alpha + \cos \beta + \cos \gamma )} \over {\cos \alpha \cos \beta \cos \gamma }} = 12$
then, ${\left( {{{\cos 3\alpha + \cos 3\beta + \cos 3\gamma } \over {\cos \alpha \cos \beta \cos \gamma }}} \right)^2} = 144$
2021
Q106
JEE Mains
Numerical
14 Mar 2026
The maximum value of z in the following equation z = 6xy + y2, where 3x + 4y $ \le $ 100 and 4x + 3y $ \le $ 75 for x $ \ge $ 0 and y $ \ge $ 0 is __________.
Correct Answer: 904
Explanation:
3x + 4y $ \le $ 100
4x + 3y $ \le $ 75
x $ \ge $ 0, y $ \ge $ 0
Feasible region is shown in the graph
Let maximum value of 6xy + y2 = c
For a solution with feasible region,
6xy + y2 = c and 4x + 3y = 75 must have at least one positive solution.
${y^2} + 6y\left( {{{75 - 3y} \over 4}} \right) - c = 0 $
$\Rightarrow {7 \over 2}{y^2} - {{225} \over 2}y + c = 0$
$ \Rightarrow {\left( {{{225} \over 2}} \right)^2} \ge 4.{7 \over 2}.c $
$\Rightarrow c \le {{{{225}^2}} \over {56}} \approx 904$
2020
Q107
JEE Mains
MCQ
14 Mar 2026
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.
$\left( {{{11} \over 5},{{28} \over 5}} \right)$
B.
$\left( {{{29} \over 5},{{11} \over 5}} \right)$
C.
$\left( {{{29} \over 5},{8 \over 5}} \right)$
D.
$\left( {{8 \over 5},{{29} \over 5}} \right)$
2020
Q108
JEE Mains
MCQ
14 Mar 2026
A ray of light coming from the point (2, $2\sqrt 3 $) is incident at an angle 30o on the line x = 1 at the
point A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB
passes through the point :
A.
(3, -$\sqrt 3 $)
B.
(4, -$\sqrt 3 $)
C.
$\left( {4, - {{\sqrt 3 } \over 2}} \right)$
D.
$\left( {3, - {1 \over {\sqrt 3 }}} \right)$
2020
Q109
JEE Mains
MCQ
14 Mar 2026
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.
-4
C.
–2
D.
$\sqrt {15} $
2020
Q110
JEE Mains
MCQ
14 Mar 2026
Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A
and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m)
above the line AC is :
A.
10/3
B.
5
C.
20/3
D.
6
2020
Q111
JEE Mains
MCQ
14 Mar 2026
If a $\Delta $ABC has vertices A(–1, 7), B(–7, 1) and
C(5, –5), then its orthocentre has coordinates :
A.
(–3, 3)
B.
(3, –3)
C.
$\left( {{3 \over 5}, - {3 \over 5}} \right)$
D.
$\left( { - {3 \over 5},{3 \over 5}} \right)$
2020
Q112
JEE Mains
MCQ
14 Mar 2026
The set of all possible values of
$\theta $ in the interval
(0, $\pi $) for which the points (1, 2) and (sin $\theta $, cos $\theta $) lie
on the same side of the line x + y = 1 is :
(0, $\pi $) for which the points (1, 2) and (sin $\theta $, cos $\theta $) lie
on the same side of the line x + y = 1 is :
A.
$\left( {0,{\pi \over 4}} \right)$
B.
$\left( {0,{{3\pi } \over 4}} \right)$
C.
$\left( {{\pi \over 4},{{3\pi } \over 4}} \right)$
D.
$\left( {0,{\pi \over 2}} \right)$
2020
Q113
JEE Mains
MCQ
14 Mar 2026
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, 7)
C.
(7, 6)
D.
(–9, –6)
2020
Q114
JEE Mains
MCQ
14 Mar 2026
Let two points be A(1, –1) and B(0, 2). If a point
P(x', y') be such that the area of $\Delta $PAB = 5 sq.
units and it lies on the line, 3x + y – 4$\lambda $ = 0,
then a value of $\lambda $ is :
A.
4
B.
1
C.
-3
D.
3
2020
Q115
JEE Mains
MCQ
14 Mar 2026
The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line
x = y is :
A.
3x - 2y = 0
B.
7x - 5y = 0
C.
2x - 3y = 0
D.
5x - 7y = 0
2020
Q116
JEE Mains
Numerical
14 Mar 2026
If the line, 2x - y + 3 = 0 is at a distance
${1 \over {\sqrt 5 }}$ and ${2 \over {\sqrt 5 }}$ 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 :
${1 \over {\sqrt 5 }}$ and ${2 \over {\sqrt 5 }}$ 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 :
Correct Answer: 30
Explanation:
Apply distance between parallel line formula
$4x - 2y + \alpha = 0$
$4x - 2y + 6 = 0$
$\left| {{{\alpha - 6} \over {25}}} \right| = {1 \over {55}}$
$|\alpha - 6|\, = 2 \Rightarrow \alpha = 8,4$
sum = 12
Again
$6x - 3y + \beta = 0$
$6x - 3y + 9 = 0$
$\left| {{{\beta - 9} \over {3\sqrt 5 }}} \right| = {2 \over {\sqrt 5 }}$
$|\beta - 9|\, = 6 \Rightarrow \beta = 15,3$
sum = 18
$ \therefore $ Sum of all values of $\alpha$ and $\beta$ is = 30
$4x - 2y + \alpha = 0$
$4x - 2y + 6 = 0$
$\left| {{{\alpha - 6} \over {25}}} \right| = {1 \over {55}}$
$|\alpha - 6|\, = 2 \Rightarrow \alpha = 8,4$
sum = 12
Again
$6x - 3y + \beta = 0$
$6x - 3y + 9 = 0$
$\left| {{{\beta - 9} \over {3\sqrt 5 }}} \right| = {2 \over {\sqrt 5 }}$
$|\beta - 9|\, = 6 \Rightarrow \beta = 15,3$
sum = 18
$ \therefore $ Sum of all values of $\alpha$ and $\beta$ is = 30
2020
Q117
JEE Mains
Numerical
14 Mar 2026
Let A(1, 0), B(6, 2) and C $\left( {{3 \over 2},6} \right)$ 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 $\left( { - {7 \over 6}, - {1 \over 3}} \right)$, is ________.
Correct Answer: 5
Explanation:
P is centroid of the triangle ABC.
P = $\left( {{{1 + 6 + {3 \over 2}} \over 3},{{0 + 2 + 6} \over 3}} \right)$
= $\left( {{{17} \over 6},{8 \over 3}} \right)$
Given Q $\left( { - {7 \over 6}, - {1 \over 3}} \right)$.
$ \therefore $ PQ = $\sqrt {{{\left( {{{24} \over 6}} \right)}^2} + {{\left( {{9 \over 3}} \right)}^2}} $ = 5
2019
Q118
JEE Mains
MCQ
14 Mar 2026
A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and
the perpendicular from the origin to this line makes an angle of 60o with the line x + y = 0. Then an equation
of the line L is :
A.
x + $\sqrt 3 $y = 8
B.
$\sqrt 3 $x + y = 8
C.
( $\sqrt 3 $ + 1)x + ( $\sqrt 3 $ – 1)y = 8 $\sqrt 2 $
D.
( $\sqrt 3 $ - 1)x + ( $\sqrt 3 $ + 1)y = 8 $\sqrt 2 $
2019
Q119
JEE Mains
MCQ
14 Mar 2026
Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance
${3 \over 5}$
from the origin. Then which one of the
following points lies on any of these lines ?
A.
$\left( {{1 \over 4}, - {1 \over 3}} \right)$
B.
$\left( { - {1 \over 4},{2 \over 3}} \right)$
C.
$\left( { - {1 \over 4}, - {2 \over 3}} \right)$
D.
$\left( {{1 \over 4},{1 \over 3}} \right)$
2019
Q120
JEE Mains
MCQ
14 Mar 2026
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 side length 2$\sqrt 2 $ units
C.
square of area 16 sq. units
D.
rhombus of side length 2 units
2019
Q121
JEE Mains
MCQ
14 Mar 2026
If the two lines x + (a – 1) y = 1 and
2x + a2y = 1 (a$ \in $R – {0, 1}) are perpendicular, then
the distance of their point of intersection from the
origin is :
A.
${2 \over \sqrt5}$
B.
${\sqrt2 \over 5}$
C.
${2 \over 5}$
D.
$\sqrt{2 \over 5}$
2019
Q122
JEE Mains
MCQ
14 Mar 2026
Slope of a line passing through P(2, 3) and
intersecting the line, x + y = 7 at a distance of
4 units from P, is :
A.
${{\sqrt 7 - 1} \over {\sqrt 7 + 1}}$
B.
${{\sqrt 5 - 1} \over {\sqrt 5 + 1}}$
C.
${{1 - \sqrt 5 } \over {1 + \sqrt 5 }}$
D.
${{1 - \sqrt 7 } \over {1 + \sqrt 7 }}$
2019
Q123
JEE Mains
MCQ
14 Mar 2026
If the system of linear equations
x – 2y + kz = 1
2x + y + z = 2
3x – y – kz = 3
has a solution (x,y,z), z $ \ne $ 0, then (x,y) lies on the straight line whose equation is :
x – 2y + kz = 1
2x + y + z = 2
3x – y – kz = 3
has a solution (x,y,z), z $ \ne $ 0, then (x,y) lies on the straight line whose equation is :
A.
4x – 3y – 4 = 0
B.
3x – 4y – 1 = 0
C.
4x – 3y – 1 = 0
D.
3x – 4y – 4 = 0
2019
Q124
JEE Mains
MCQ
14 Mar 2026
Suppose that the points (h,k), (1,2) and (–3,4) lie
on the line L1
. If a line L2
passing through the points
(h,k) and (4,3) is perpendicular to L1
, then
$k \over h$
equals :
A.
${1 \over 3}$
B.
3
C.
0
D.
-${1 \over 7}$
2019
Q125
JEE Mains
MCQ
14 Mar 2026
A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only
in :
A.
1st and 2nd qudratants
B.
4th qudratant
C.
1st and 2nd and 4th qudratants
D.
1st qudratant
2019
Q126
JEE Mains
MCQ
14 Mar 2026
Let O(0, 0) and A(0, 1) be two fixed points. Then
the locus of a point P such that the perimeter of
$\Delta $AOP is 4, is :
A.
9x2 + 8y2 – 8y = 16
B.
8x2 – 9y2 + 9y = 18
C.
8x2 + 9y2 – 9y = 18
D.
9x2 – 8y2 + 8y = 16
2019
Q127
JEE Mains
MCQ
14 Mar 2026
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.
x – y + 7 = 0
B.
4x – 3y + 24 = 0
C.
4x + 3y = 0
D.
3x – 4y + 25 = 0
2019
Q128
JEE Mains
MCQ
14 Mar 2026
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.
${{35} \over 3}$
B.
$-$ 5
C.
$-$ ${{35} \over 3}$
D.
5
2019
Q129
JEE Mains
MCQ
14 Mar 2026
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 – 11 = 0
B.
5x – 3y + 1 = 0
C.
3x – 5y + 7 = 0
D.
3x + 5y – 13 = 0
2019
Q130
JEE Mains
MCQ
14 Mar 2026
Two vertices of a triangle are (0, 2) and (4, 3). If its orthocenter is at the origin, then its third vertex lies in which quadrant :
A.
third
B.
fourth
C.
second
D.
first
2019
Q131
JEE Mains
MCQ
14 Mar 2026
Two sides of a parallelogram are along the lines, x + y = 3 & x – y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is :
A.
(2, 1)
B.
(2, 6)
C.
(3, 5)
D.
(3, 6)
2019
Q132
JEE Mains
MCQ
14 Mar 2026
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 $\Delta $PQR is a line :
A.
parallel to y-axis
B.
with slope ${2 \over 3}$
C.
parallel to x-axis
D.
with slope ${3 \over 2}$
2019
Q133
JEE Mains
MCQ
14 Mar 2026
If 5, 5r, 5r2 are the lengths of the sides of a triangle, then r cannot be equal to :
A.
${7 \over 4}$
B.
${5 \over 4}$
C.
${3 \over 4}$
D.
${3 \over 2}$
2019
Q134
JEE Mains
MCQ
14 Mar 2026
If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is :
A.
(3, 4)
B.
(2, 2)
C.
(4, 4)
D.
(4, 3)
2019
Q135
JEE Mains
MCQ
14 Mar 2026
Let the equations 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.
122y + 26x + 1675 = 0
C.
26x + 61y + 1675 = 0
D.
26x $-$ 122y $-$ 1675 = 0
2019
Q136
JEE Mains
MCQ
14 Mar 2026
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 not concurrent
B.
The lines are concurrent at the point $\left( {{3 \over 4},{1 \over 2}} \right)$
C.
The lines are all parallel
D.
Each line passes through the origin
2018
Q137
JEE Mains
MCQ
14 Mar 2026
A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is
the origin and the rectangle OPRQ is completed, then the locus of R is :
A.
3x + 2y = 6xy
B.
3x + 2y = 6
C.
2x + 3y = xy
D.
3x + 2y = xy
2018
Q138
JEE Mains
MCQ
14 Mar 2026
The foot of the perpendicular drawn from the origin, on the line, 3x + y = $\lambda $ ($\lambda $ $ \ne $ 0) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is :
A.
1 : 3
B.
3 : 1
C.
1 : 9
D.
9 : 1
2018
Q139
JEE Mains
MCQ
14 Mar 2026
The sides of a rhombus ABCD are parallel to the lines, x $-$ y + 2 = 0 and 7x $-$ y + 3 = 0. If the diagonals of the rhombus intersect P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the coordinate of A is :
A.
${5 \over 2}$
B.
${7 \over 4}$
C.
2
D.
${7 \over 2}$
2018
Q140
JEE Mains
MCQ
14 Mar 2026
In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, x + y = 5 and x = 4. Then area of $\Delta $ ABC (in sq. units) is :
A.
12
B.
4
C.
5
D.
9
2017
Q141
JEE Mains
MCQ
14 Mar 2026
A square, of each side 2, lies above the x-axis and has one vertex at the origin. If
one of the sides passing through the origin makes an angle 30o with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is :
A.
$2\sqrt 3 - 1$
B.
$2\sqrt 3 - 2$
C.
$\sqrt 3 - 2$
D.
$\sqrt 3 - 1$
2017
Q142
JEE Mains
MCQ
14 Mar 2026
Let k be an integer such that the triangle with vertices (k, – 3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :
A.
$\left( {1,{3 \over 4}} \right)$
B.
$\left( {1, - {3 \over 4}} \right)$
C.
$\left( {2,{1 \over 2}} \right)$
D.
$\left( {2, - {1 \over 2}} \right)$
2016
Q143
JEE Mains
MCQ
14 Mar 2026
A ray of light is incident along a line which meets another line, 7x − y + 1 = 0, at the point (0, 1). The ray is then reflected from this point along the line, y + 2x = 1. Then the equation of the line of incidence of the ray of light is :
A.
41x − 38y + 38 = 0
B.
41x + 25y − 25 = 0
C.
41x + 38y − 38 = 0
D.
41x − 25y + 25 = 0
2016
Q144
JEE Mains
MCQ
14 Mar 2026
A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio :
A.
2 : 3
B.
1 : 2
C.
4 : 1
D.
3 : 4
2016
Q145
JEE Mains
MCQ
14 Mar 2026
The point (2, 1) is translated parallel to the line L : x− y = 4 by $2\sqrt 3 $ units. If the newpoint Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is :
A.
x + y = 2 $-$ $\sqrt 6 $
B.
x + y = 3 $-$ 3$\sqrt 6 $
C.
x + y = 3 $-$ 2$\sqrt 6 $
D.
2x + 2y = 1 $-$ $\sqrt 6 $
2016
Q146
JEE Mains
MCQ
14 Mar 2026
If a variable line drawn through the intersection of the lines ${x \over 3} + {y \over 4} = 1$ and ${x \over 4} + {y \over 3} = 1,$ meets the coordinate axes at A and B, (A $ \ne $ B), then the locus of the midpoint of AB is :
A.
6xy = 7(x + y)
B.
4(x + y)2 − 28(x + y) + 49 = 0
C.
7xy = 6(x + y)
D.
14(x + y)2 − 97(x + y) + 168 = 0
2016
Q147
JEE Mains
MCQ
14 Mar 2026
Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7x - y - 5 = 0$. If its diagonals intersect at $(-1, -2)$, then which one of the following is a vertex of this rhombus?
A.
$\left( {{{ 1} \over 3}, - {8 \over 3}} \right)$
B.
$\left( - {{{ 10} \over 3}, - {7 \over 3}} \right)$
C.
$\left( { - 3, - 9} \right)$
D.
$\left( { - 3, - 8} \right)$
2015
Q148
JEE Mains
MCQ
14 Mar 2026
The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0)$ $(0, 41)$ and $(41, 0)$ is :
A.
820
B.
780
C.
901
D.
861
2014
Q149
JEE Mains
MCQ
14 Mar 2026
Let $a, b, c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax + 2ay + c = 0$ and $5bx + 2by + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then :
A.
$3bc - 2ad = 0$
B.
$3bc + 2ad = 0$
C.
$2bc - 3ad = 0$
D.
$2bc + 3ad = 0$
2014
Q150
JEE Mains
MCQ
14 Mar 2026
Let $PS$ be the median of the triangle with vertices $P(2, 2)$, $Q(6, -1)$ and $R(7, 3)$. The equation of the line passing through $(1, -1)$ band parallel to PS is :
A.
$4x + 7y + 3 = 0$
B.
$2x - 9y - 11 = 0$
C.
$4x - 7y - 11 = 0$
D.
$2x + 9y + 7 = 0$
