Straight Lines and Pair of Straight Lines
576 Questions
Start JEE Mains Test
2003
Q501
JEE Mains
MCQ
14 Mar 2026
If the pair of straight lines ${x^2} - 2pxy - {y^2} = 0$ and ${x^2} - 2qxy - {y^2} = 0$ be such that each pair bisects the angle between the other pair, then :
A.
$pq = -1$
B.
$p = q$
C.
$p = -q$
D.
$pq = 1$.
2003
Q502
JEE Mains
MCQ
14 Mar 2026
If ${x_1},{x_2},{x_3}$ and ${y_1},{y_2},{y_3}$ are both in G.P. with the same common ratio, then the points $\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)$ and $\left( {{x_3},{y_3}} \right)$ :
A.
are vertices of a triangle
B.
lie on a straight line
C.
lie on an ellipse
D.
lie on a circle
2003
Q503
JEE Mains
MCQ
14 Mar 2026
A square of side a lies above the $x$-axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha \left( {0 < \alpha < {\pi \over 4}} \right)$ with the positive direction of x-axis. The equation of its diagonal not passing through the origin is :
A.
$y\left( {\cos \alpha + \sin \alpha } \right) + x\left( {\cos \alpha - \sin \alpha } \right) = a$
B.
$y\left( {\cos \alpha - \sin \alpha } \right) - x\left( {\sin \alpha - \cos \alpha } \right) = a$
C.
$y\left( {\cos \alpha + \sin \alpha } \right) + x\left( {\sin \alpha - \cos \alpha } \right) = a$
D.
$y\left( {\cos \alpha + \sin \alpha } \right) + x\left( {\sin \alpha + \cos \alpha } \right) = a$
2003
Q504
JEE Advanced
MCQ
14 Mar 2026
The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices $\left( {0,0} \right),\left( {0,21} \right)$ and $\left( {21,0} \right)$, is
A.
133
B.
190
C.
233
D.
105
2003
Q505
JEE Advanced
MCQ
14 Mar 2026
Orthocentre of triangle with vertices $\left( {0,0} \right),\left( {3,4} \right)$ and $\left( {4,0} \right)$ is
A.
$\,\,\left( {3,{5 \over 4}} \right)$
B.
$\left( {3,12} \right)$
C.
$\left( {3,{3 \over 4}} \right)$
D.
$\left( {3,9} \right)$
2002
Q506
JEE Mains
MCQ
14 Mar 2026
If the pair of lines
$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$
intersect on the $y$-axis then :
$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$
intersect on the $y$-axis then :
A.
$2fgh = b{g^2} + c{h^2}$
B.
$b{g^2} \ne c{h^2}$
C.
$abc = 2fgh$
D.
none of these
2002
Q507
JEE Mains
MCQ
14 Mar 2026
The pair of lines represented by
$$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$
are perpendicular to each other for :
are perpendicular to each other for :
A.
two values of $a$
B.
$\forall \,a$
C.
for one value of $a$
D.
for no values of $a$
2002
Q508
JEE Mains
MCQ
14 Mar 2026
Locus of mid point of the portion between the axes of
$x$ $cos$ $\alpha + y\,\sin \alpha = p$ where $p$ is constant is :
$x$ $cos$ $\alpha + y\,\sin \alpha = p$ where $p$ is constant is :
A.
${x^2} + {y^2} = {4 \over {{p^2}}}$
B.
${x^2} + {y^2} = 4{p^2}$
C.
${1 \over {{x^2}}} + {1 \over {{y^2}}} = {2 \over {{p^2}}}$
D.
${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}$
2002
Q509
JEE Mains
MCQ
14 Mar 2026
A triangle with vertices $\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)$ is :
A.
isosceles and right angled
B.
isosceles but not right angled
C.
right angled but not isosceles
D.
neither right angled nor isosceles
2002
Q510
JEE Advanced
MCQ
14 Mar 2026
Let $0 < \alpha < {\pi \over 2}$ be fixed angle. If $P = \left( {\cos \theta ,\,\sin \theta } \right)$ and $Q = \left( {\cos \left( {\alpha - \theta } \right),\,\sin \left( {\alpha - \theta } \right)} \right),$ then $Q$ is obtained from $P$ by
A.
clockwise rotation around origin through an angle $\alpha $
B.
anticlockwise rotation around origin through an angle $\alpha $
C.
reflection in the line through origin with slope tan $\alpha $
D.
reflection in the line through origin with slope tan $\left( {\alpha /2} \right)$
2002
Q511
JEE Advanced
MCQ
14 Mar 2026
A straight line through the origin $O$ meets the parallel lines $4x+2y=9$ and $2x+y+6=0$ at points $P$ and $Q$ respectively. Then the point $O$ divides the segemnt $PQ$ in the ratio
A.
$1 : 2$
B.
$3 : 4$
C.
$2 : 1$
D.
$4 : 3$
2002
Q512
JEE Advanced
MCQ
14 Mar 2026
Let $P = \left( { - 1,\,0} \right),\,Q = \left( {0,\,0} \right)$ and $R = \left( {3,\,3\sqrt 3 } \right)$ be three points.
Then the equation of the bisector of the angle $PQR$ is
Then the equation of the bisector of the angle $PQR$ is
A.
${{\sqrt 3 } \over 2}x + y = 0$
B.
$x + \sqrt 3 y = 0$
C.
$\sqrt 3 x + y = 0$
D.
$x + {{\sqrt 3 } \over 2}y = 0$
2002
Q513
JEE Advanced
MCQ
14 Mar 2026
If the pair of lines $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$ intersect on the $y$ axis then
A.
$2fgh = b{g^2} + c{h^2}$
B.
$b{g^2} \ne c{h^2}$
C.
$\,abc = 2fgh$
D.
none of these
2002
Q514
JEE Advanced
MCQ
14 Mar 2026
A triangle with vertices $(4, 0), (-1, -1), (3, 5)$is
A.
isosceles and right angled
B.
isosceles but not right angled
C.
right angled but not isosceles
D.
neither right angled nor isosceles
2002
Q515
JEE Advanced
MCQ
14 Mar 2026
Locus of mid point of the portion between the axes of $x$ $\cos \alpha + y\sin \alpha = p$ where $p$ is constant is
A.
${x^2} + {y^2} = {4 \over {{p^2}}}\,\,\,$
B.
${x^2} + {y^2} = 4{p^2}$
C.
${1 \over {{x^2}}} + {1 \over {{y^2}}} = {2 \over {{p^2}}}$
D.
${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}$
2002
Q516
JEE Advanced
MCQ
14 Mar 2026
The pair of lines represented by
$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$ are perpendicular to each other for
$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$ are perpendicular to each other for
A.
two values of $a$
B.
$\forall \,a$
C.
for one values of $a$
D.
for no values of $a$
2002
Q517
JEE Advanced
Numerical
14 Mar 2026
A straight line $L$ through the origin meets the lines $x + y = 1$ and $x + y = 3$ at $P $ and $Q$ respectively. Through $P$ and $Q$ two straight lines ${L_1}$ and ${L_2}$ are drawn, parallel to $2x - y = 5$ and $3x + y = 5$ respectively. Lines ${L_1}$ and ${L_2}$ intersect at $R$. Show that the locus of $R$, as $L$ varies is a straight line.
Correct Answer: Solve it.
2002
Q518
JEE Advanced
Numerical
14 Mar 2026
A straight line $L$ with negative slope passes through the point $(8, 2)$ and cuts the positive coordinate axes at points $P$ and $Q$. Find the absolute minimum value of $OP + OQ,$ as $L$ varies, where $O$ is the origin.
Correct Answer: 18
2001
Q519
JEE Advanced
MCQ
14 Mar 2026
The number of integer values of $m$, for which the $x$-coordinate of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is also an integer, is
A.
2
B.
0
C.
4
D.
1
2001
Q520
JEE Advanced
MCQ
14 Mar 2026
Area of the parallelogram formed by the lines $y = mx$, $y = mx + 1$, $y = nx$ and $y = nx + 1$ equals
A.
$\left| {m + n} \right|/{\left( {m - n} \right)^2}$
B.
$2/\left| {m + n} \right|$
C.
$1/\left( {\left| {m + n} \right|} \right)$
D.
$1/\left( {\left| {m - n} \right|} \right)$
2001
Q521
JEE Advanced
Numerical
14 Mar 2026
Let $a, b, c$ be real numbers with ${a^2} + {b^2} + {c^2} = 1.$ Show that
represents a straight line.
the equation $\left| {\matrix{ {ax - by - c} & {bx + ay} & {cx + a} \cr {bx + ay} & { - ax + by - c} & {cy + b} \cr {cx + a} & {cy + b} & { - ax - by + c} \cr } } \right| = 0$
represents a straight line.
Correct Answer: Solve it.
2000
Q522
JEE Advanced
MCQ
14 Mar 2026
The incentre of the triangle with vertices $\left( {1,\,\sqrt 3 } \right),\left( {0,\,0} \right)$ and $\left( {2,\,0} \right)$ is
A.
$\left( {1,\,{{\sqrt 3 } \over 2}} \right)$
B.
$\left( {{2 \over 3},\,{1 \over {\sqrt 3 }}} \right)$
C.
$\left( {{2 \over 3},\,{{\sqrt 3 } \over 2}} \right)$
D.
$\left( {1,\,{1 \over {\sqrt 3 }}} \right)$
2000
Q523
JEE Advanced
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 $(1, -1)$ and parallel to $PS$ is
A.
$2x - 9y - 7 = 0$
B.
$2x - 9y - 11 = 0$
C.
$2x + 9y - 11 = 0$
D.
$2x + 9y + 7 = 0$
2000
Q524
JEE Advanced
Numerical
14 Mar 2026
For points $P\,\,\, = \left( {{x_1},\,{y_1}} \right)$ and $Q\,\,\, = \left( {{x_2},\,{y_2}} \right)$ of the co-ordinate plane, a new distance $d\left( {P,\,Q} \right)$ is defined by $d\left( {P,\,Q} \right)$$ = \left( {{x_2},\,{y_2}} \right)\left| {{x_1} - {x_2}} \right| + \left| {{y_1} - {y_2}} \right|.$ Let $O = (0, 0)$ and $A = (3, 2)$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $O$ and $A$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.
Correct Answer: Solve it.
2000
Q525
JEE Advanced
Numerical
14 Mar 2026
Let $ABC$ and $PQR$ be any two triangles in the same plane. Assume that the prependiculars from the points $A, B, C$ to the sides $QR, RP, PQ$ respectively are concurrent. Using vector methods or otherwise, prove that the prependiculars from $P, Q, R $ to $BC,$ $CA$, $AB$ respectively are also concurrent.
Correct Answer: Solve it.
1999
Q526
JEE Advanced
MCQ
14 Mar 2026
If ${x_1},\,{x_2},\,{x_3}$ as well as ${y_1},\,{y_2},\,{y_3}$, are in G.P. with the same common ratio, then the points $\left( {{x_1},\,{y_1}} \right),\left( {{x_2},\,{y_2}} \right)$ and $\left( {{x_3},\,{y_3}} \right).$
A.
lie on a straight line
B.
lie on an ellipse
C.
lie on a circle
D.
are vertices of a triangle
1999
Q527
JEE Advanced
MCQ
14 Mar 2026
Lt $PQR$ be a right angled isosceles triangle, right angled at $P(2, 1)$. If the equation of the line $QR$ is $2x + y = 3,$ then the equation representing the pair of lines $PQ$ and $PR$ is
A.
$3{x^2} - 3{y^2} + 8xy + 20x + 10y + 25 = 0$
B.
$3{x^2} - 3{y^2} + 8xy - 20x - 10y + 25 = 0$
C.
$3{x^2} - 3{y^2} + 8xy + 10x + 15y + 20 = 0$
D.
$3{x^2} - 3{y^2} - 8xy - 10x - 15y - 20 = 0$
1999
Q528
JEE Advanced
MSQ
14 Mar 2026
Let ${L_1}$ be a straight line passing through the origin and ${L_2}$ be the straight line $x + y = 1$. If the intercepts made by the circle ${x^2} + {y^2} - x + 3y = 0$ on ${L_1}$ and ${L_2}$ are equal, then which of the following equations can represent ${L_1}$?
A.
$x + y = 0$
B.
$x -y = 0$
C.
$x + 7y = 0$
D.
$x - 7y = 0$
1998
Q529
JEE Advanced
MCQ
14 Mar 2026
The diagonals of a parralleogram $PQRS$ are along the lines $x + 3y = 4$ and $6x - 2y = 7$. Then $PQRS$ must be a.
A.
rectangle
B.
square
C.
cyclic quadrilateral
D.
rhombus.
1998
Q530
JEE Advanced
MCQ
14 Mar 2026
If $\left( {P\left( {1,2} \right),\,Q\left( {4,6} \right),\,R\left( {5,7} \right)} \right)$ and $S\left( {a,b} \right)$ are the vertices of a parrallelogram $PQRS,$ then
A.
$a = 2,\,b = 4$
B.
$a = 3,\,b = 4$
C.
$a = 2,\,b = 3$
D.
$a = 3,\,b = 5$
1998
Q531
JEE Advanced
MSQ
14 Mar 2026
If the vertices $P, Q, R$ of a triangle $PQR$ are rational points, which of the following points of the triangle $PQR$ is (are) always rational point(s)?
A.
centroid ( A rational point is a point both of whose co-ordinates are rational numbers.)
B.
incentre. ( A rational point is a point both of whose co-ordinates are rational numbers.)
C.
circumcentre ( A rational point is a point both of whose co-ordinates are rational numbers.)
D.
orthocentre ( A rational point is a point both of whose co-ordinates are rational numbers.)
1998
Q532
JEE Advanced
Numerical
14 Mar 2026
Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.
Correct Answer: Solve it.
1996
Q533
JEE Advanced
Numerical
14 Mar 2026
A rectangle $PQRS$ has its side $PQ$ parallel to the line $y = mx$ and vertices $P, Q$ and $S$ on the lines $y = a, x = b$ and $x = -b,$ respectively. Find the locus of the vertex $R$.
Correct Answer: $$x\left( {{m^2} - 1} \right) - ym + \left( {{m^2} + 1} \right)b + am = 0$$
1995
Q534
JEE Advanced
MCQ
14 Mar 2026
The orthocentre of the triangle formed by the lines $xy=0$ and $x+y=1$ is
A.
$\left( {{1 \over 2},\,{1 \over 2}} \right)$
B.
$\left( {{1 \over 3},\,{1 \over 3}} \right)$
C.
$\left( {0,\,0} \right)$
D.
$\left( {{1 \over 4},\,{1 \over 4}} \right)$
1994
Q535
JEE Advanced
MCQ
14 Mar 2026
The locus of a variable point whose distance from $\left( { - 2,\,0} \right)$ is $2/3$ times its distance from the line $x = - {9 \over 2}$ is
A.
ellipse -
B.
parabola
C.
hyperbola
D.
none of these
1994
Q536
JEE Advanced
MCQ
14 Mar 2026
The equations to a pair of opposites sides of parallelogram are ${x^2} - 5x + 6 = 0$ and ${y^2} - 6y + 5 = 0,$ the equations to its diagonals are
A.
$x + 4y = 13,\,y = 4x - 7$
B.
$4x + y = 13,\,4y = x - 7$
C.
$4x + y = 13,\,y = 4x - 7$
D.
$y - 4x = 13,\,y + 4x = 7$
1993
Q537
JEE Advanced
Numerical
14 Mar 2026
A line through $A (-5, -4)$ meets the line $x + 3y + 2 = 0,$ $2x + y + 4 = 0$ and $x - y - 5 = 0$ at the points $B, C$ and $D$ respectively. If ${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)^2},$ find the equation of the line.
Correct Answer: $$2x + 3y + 22 = 0$$
1993
Q538
JEE Advanced
Numerical
14 Mar 2026
Tagent at a point ${P_1}$ {other than $(0, 0)$} on the curve $y = {x^3}$ meets the curve again at ${P_2}$. The tangent at ${P_2}$ meets the curve at ${P_3}$, and so on. Show that the abscissae of ${P_1},\,{P_2},{P_3}......{P_n},$ form a G.P. Also find the ratio.
[area $\left( {\Delta {P_1},{P_2},{P_3}} \right)$]/[area $\left( {{P_2},{P_3},{P_4}} \right)$]
Correct Answer: $${1 \over {64}}\,sq.\,unit$$
1993
Q539
JEE Advanced
Numerical
14 Mar 2026
The vertices of a triangle are $A\left( { - 1, - 7} \right)B\left( {5,\,1} \right)$ and $C\left( {1,\,4} \right).$ The equation of the bisector of the angle $\angle ABC$ is ............... .
Correct Answer: $$x - 7y + 2 = 0$$
1992
Q540
JEE Advanced
MCQ
14 Mar 2026
If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
A.
square
B.
circle
C.
straight line
D.
two intersecting lines
1992
Q541
JEE Advanced
Numerical
14 Mar 2026
Determine all values of $\alpha $ for which the point $\left( {\alpha ,\,{\alpha ^2}} \right)$ lies insides the triangle formed by the lines
$$\matrix{
{2x + 3y - 1 = 0} \cr
{x + 2y - 3 = 0} \cr
{5x - 6y - 1 = 0} \cr
} $$
Correct Answer: $$\alpha \in \left( { - {3 \over 2}, - 1} \right) \cup \left( {{1 \over 2},1} \right)$$
1991
Q542
JEE Advanced
Numerical
14 Mar 2026
Find the equation of the line passing through the point $(2, 3)$ and making intercept of length 2 units between the lines $y + 2x = 3$ and $y + 2x = 5$.
Correct Answer: $$3x + 4y - 18 = 0$$ or $$x - 2 = 0$$
1991
Q543
JEE Advanced
Numerical
14 Mar 2026
Show that all chords of the curve $3{x^2} - {y^2} - 2x + 4y = 0,$ which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.
Correct Answer: $$(1, -2)$$
1991
Q544
JEE Advanced
Numerical
14 Mar 2026
Let the algebraic sum of the perpendicular distances from the points $\left( {2,0} \right),\,\left( {0,\,2} \right)$ $\left( {1,\,1} \right)$ to a variable straight line be zero; then the line passes through a fixed point whose cordinates are ...............
Correct Answer: $$\left( {1,\,1} \right)$$
1990
Q545
JEE Advanced
MCQ
14 Mar 2026
Line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$, then
A.
${a^2} + {b^2} = {p^2} + {q^2}$
B.
${1 \over {{a^2}}} + {1 \over {{b^2}}} = {1 \over {{p^2}}} + {1 \over {{q^2}}}$
C.
${a^2} + {p^2} = {b^2} + {q^2}$
D.
${1 \over {{a^2}}} + {1 \over {{p^2}}} = {1 \over {{b^2}}} + {1 \over {{q^2}}}$
1990
Q546
JEE Advanced
Numerical
14 Mar 2026
Straight lines $3x + 4y = 5$ and $4x - 3y = 15$ intersect at the point $A$. Points $B$ and $C$ are choosen on these two lines such that $AB = AC$. Determine the possible equations of the line $BC$ passing through the point $(1, 2)$.
Correct Answer: $$x - 7y + 13 = 0$$ or $$7x + y - 9 = 0$$
1990
Q547
JEE Advanced
Numerical
14 Mar 2026
A line cuts the $x$-axis at $A (7, 0)$ and the $y$-axis at $B (0, -5)$. A variable line $PQ$ is drawn perpendicular to $AB$ cutting the $x$axis in $P$ and they $Y$-axis in $Q$. If $AQ$ and $BP$ intersect at $R$, find the locus of R.
Correct Answer: $${x^2} + {y^2} - 7x + 5y = 0$$
1989
Q548
JEE Advanced
Numerical
14 Mar 2026
Let $ABC$ be a triangle with $AB = AC$. If $D$ is the midpoint of $BC, E$ is the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that $AF$ is perpendicular to $BE$.
Correct Answer: Solve it.
1988
Q549
JEE Advanced
MCQ
14 Mar 2026
If $P=(1, 0),$ $Q=(-1, 0)$ and $R=(2, 0)$ are three given points, then locus of the point $S$ satisfying the relation $S{Q^2} + S{R^2} = 2S{P^2},$ is
A.
a straight line parallel to x-axis
B.
a circle passing through the origin
C.
a circle with the centre at the origin
D.
a straight line parallel to y-axis.
1988
Q550
JEE Advanced
Numerical
14 Mar 2026
Lines${L_1} = ax + by + c = 0$ and ${L_2} = lx + my + n = 0$ intersect at the point $P$ and make an angle $\theta $ with each other. Find the equation of a line $L$ different from ${L_2}$ which passes through $P$ and makes the same angle $\theta $ with ${L_1}$.
Correct Answer: $$\left( {{a^2} + {b^2}} \right)\left( {\ell x + my + n} \right) - 2\left( {a\ell + bm} \right)\left( {ax + by + c} \right) = 0$$