← Back to Mathematics

Straight Lines and Pair of Straight Lines

172 Questions
All Exams JEE Mains JEE Advanced
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Evening Slot
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 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Evening Slot
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 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Evening Slot
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 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Morning Slot
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 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Morning Slot
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 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Morning Slot
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 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Evening Slot
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 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Morning Slot
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 JEE Mains MCQ
JEE Main 2018 (Offline)
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 JEE Mains MCQ
JEE Main 2018 (Online) 15th April Evening Slot
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 JEE Mains MCQ
JEE Main 2018 (Online) 15th April Evening Slot
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 JEE Mains MCQ
JEE Main 2018 (Online) 15th April Morning Slot
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 JEE Mains MCQ
JEE Main 2017 (Online) 9th April Morning Slot
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 JEE Mains MCQ
JEE Main 2017 (Offline)
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 JEE Mains MCQ
JEE Main 2016 (Online) 10th April Morning Slot
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 JEE Mains MCQ
JEE Main 2016 (Online) 10th April Morning Slot
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 JEE Mains MCQ
JEE Main 2016 (Online) 9th April Morning Slot
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 JEE Mains MCQ
JEE Main 2016 (Online) 9th April Morning Slot
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 JEE Mains MCQ
JEE Main 2016 (Offline)
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 JEE Mains MCQ
JEE Main 2015 (Offline)
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 JEE Mains MCQ
JEE Main 2014 (Offline)
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 JEE Mains MCQ
JEE Main 2014 (Offline)
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.
$2x + 9y + 7 = 0$
2013 JEE Mains MCQ
JEE Main 2013 (Offline)
A ray of light along $x + \sqrt 3 y = \sqrt 3 $ gets reflected upon reaching $X$-axis, the equation of the reflected ray is :
A.
$y = x + \sqrt 3 $
B.
$\sqrt 3 y = x - \sqrt 3 $
C.
$y = \sqrt 3 x - \sqrt 3 $
D.
$\sqrt 3 y = x - 1$
2013 JEE Mains MCQ
JEE Main 2013 (Offline)
The $x$-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $(0, 1) (1, 1)$ and $(1, 0)$ is :
A.
$2 + \sqrt 2 $
B.
$2 - \sqrt 2 $
C.
$1 + \sqrt 2 $
D.
$1 - \sqrt 2 $
2012 JEE Mains MCQ
AIEEE 2012
If the line $2x + y = k$ passes through the point which divides the line segment joining the points $(1, 1)$ and $(2, 4)$ in the ratio $3 : 2$, then $k$ equals :
A.
${{29 \over 5}}$
B.
$5$
C.
$6$
D.
${{11 \over 5}}$
2011 JEE Mains MCQ
AIEEE 2011
The lines ${L_1}:y - x = 0$ and ${L_2}:2x + y = 0$ intersect the line ${L_3}:y + 2 = 0$ at $P$ and $Q$ respectively. The bisector of the acute angle between ${L_1}$ and ${L_2}$ intersects ${L_3}$ at $R$.

Statement-1: The ratio $PR$ : $RQ$ equals $2\sqrt 2 :\sqrt 5 $
Statement-2: In any triangle, bisector of an angle divide the triangle into two similar triangles.

A.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B.
Statement-1 is true, Statement-2 is false.
C.
Statement-1 is false, Statement-2 is true.
D.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
2010 JEE Mains MCQ
AIEEE 2010
The line $L$ given by ${x \over 5} + {y \over b} = 1$ passes through the point $\left( {13,32} \right)$. The line K is parrallel to $L$ and has the equation ${x \over c} + {y \over 3} = 1.$ Then the distance between $L$ and $K$ is :
A.
$\sqrt {17} $
B.
${{17} \over {\sqrt {15} }}$
C.
${{23} \over {\sqrt {17} }}$
D.
${{23} \over {\sqrt {15} }}$
2009 JEE Mains MCQ
AIEEE 2009
The shortest distance between the line $y - x = 1$ and the curve $x = {y^2}$ is :
A.
${{2\sqrt 3 } \over 8}$
B.
${{3\sqrt 2 } \over 5}$
C.
${{\sqrt 3 } \over 4}$
D.
${{3\sqrt 2 } \over 8}$
2009 JEE Mains MCQ
AIEEE 2009
The lines $p\left( {{p^2} + 1} \right)x - y + q = 0$ and $\left( {{p^2} + 1} \right){}^2x + \left( {{p^2} + 1} \right)y + 2q$ $=0$ are perpendicular to a common line for :
A.
exactly one values of $p$
B.
exactly two values of $p$
C.
more than two values of $p$
D.
no value of $p$
2008 JEE Mains MCQ
AIEEE 2008
The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is :
A.
1
B.
2
C.
-2
D.
-4
2007 JEE Mains MCQ
AIEEE 2007
Let A $\left( {h,k} \right)$, B$\left( {1,1} \right)$ and C $(2, 1)$ be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is $1$ square unit, then the set of values which $'k'$ can take is given by :
A.
$\left\{ { - 1,3} \right\}$
B.
$\left\{ { - 3, - 2} \right\}$
C.
$\left\{ { 1,3} \right\}$
D.
$\left\{ {0,2} \right\}$
2007 JEE Mains MCQ
AIEEE 2007
If one of the lines of $m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0$ is a bisector of angle between the lines $xy = 0,$ then $m$ is :
A.
$1$
B.
$2$
C.
$-1/2$
D.
$-2$
2007 JEE Mains MCQ
AIEEE 2007
Let $P = \left( { - 1,0} \right),\,Q = \left( {0,0} \right)$ and $R = \left( {3,3\sqrt 3 } \right)$ be three point. The equation of the bisector of the angle $PQR$ is :
A.
${{\sqrt 3 } \over 2}x + y = 0$
B.
$x + \sqrt {3y} = 0$
C.
$\sqrt 3 x + y = 0$
D.
$x + {{\sqrt 3 } \over 2}y = 0$
2006 JEE Mains MCQ
AIEEE 2006
If $\left( {a,{a^2}} \right)$ falls inside the angle made by the lines $y = {x \over 2},$ $x > 0$ and $y = 3x,$ $x > 0,$ then a belong to :
A.
$\left( {0,{1 \over 2}} \right)$
B.
$\left( {3,\infty } \right)$
C.
$\left( {{1 \over 2},3} \right)$
D.
$\left( {-3,-{1 \over 2}} \right)$
2006 JEE Mains MCQ
AIEEE 2006
A straight line through the point $A (3, 4)$ is such that its intercept between the axes is bisected at $A$. Its equation is :
A.
$x + y = 7$
B.
$3x - 4y + 7 = 0$
C.
$4x + 3y = 24$
D.
$3x + 4y = 25$
2005 JEE Mains MCQ
AIEEE 2005
If a vertex of a triangle is $(1, 1)$ and the mid points of two sides through this vertex are $(-1, 2)$ and $(3, 2)$ then the centroid of the triangle is :
A.
$\left( { - 1,{7 \over 3}} \right)$
B.
$\left( {{{ - 1} \over 3},{7 \over 3}} \right)$
C.
$\left( { 1,{7 \over 3}} \right)$
D.
$\left( {{{ 1} \over 3},{7 \over 3}} \right)$
2005 JEE Mains MCQ
AIEEE 2005
If non zero numbers $a, b, c$ are in $H.P.,$ then the straight line ${x \over a} + {y \over b} + {1 \over c} = 0$ always passes through a fixed point. That point is :
A.
$(-1,2)$
B.
$(-1, -2)$
C.
$(1, -2)$
D.
$\left( {1, - {1 \over 2}} \right)$
2005 JEE Mains MCQ
AIEEE 2005
The line parallel to the $x$ - axis and passing through the intersection of the lines $ax + 2by + 3b = 0$ and $bx - 2ay - 3a = 0,$ where $(a, b)$ $ \ne $ $(0, 0)$ is :
A.
below the $x$ - axis at a distance of ${3 \over 2}$ from it
B.
below the $x$ - axis at a distance of ${2 \over 3}$ from it
C.
above the $x$ - axis at a distance of ${3 \over 2}$ from it
D.
above the $x$ - axis at a distance of ${2 \over 3}$ from it
2004 JEE Mains MCQ
AIEEE 2004
If the sum of the slopes of the lines given by ${x^2} - 2cxy - 7{y^2} = 0$ is four times their product $c$ has the value :
A.
$-2$
B.
$-1$
C.
$2$
D.
$1$
2004 JEE Mains MCQ
AIEEE 2004
If one of the lines given by $6{x^2} - xy + 4c{y^2} = 0$ is $3x + 4y = 0,$ then $c$ equals :
A.
$-3$
B.
$-1$
C.
$3$
D.
$1$
2004 JEE Mains MCQ
AIEEE 2004
The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is :
A.
${x \over 2} - {y \over 3} = 1$ and ${x \over -2} +{y \over 1} = 1$
B.
${x \over 2} - {y \over 3} = -1$ and ${x \over -2} +{y \over 1} = -1$
C.
${x \over 2} + {y \over 3} = 1$ and ${x \over 2} +{y \over 1} = 1$
D.
${x \over 2} + {y \over 3} = -1$ and ${x \over -2} +{y \over 1} = -1$
2004 JEE Mains MCQ
AIEEE 2004
Let $A\left( {2, - 3} \right)$ and $B\left( {-2, 1} \right)$ be vertices of a triangle $ABC$. If the centroid of this triangle moves on the line $2x + 3y = 1$, then the locus of the vertex $C$ is the line :
A.
$3x - 2y = 3$
B.
$2x - 3y = 7$
C.
$3x + 2y = 5$
D.
$2x + 3y = 9$
2003 JEE Mains MCQ
AIEEE 2003
Locus of centroid of the triangle whose vertices are $\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \right)$ and $\left( {1,0} \right),$ where $t$ is a parameter, is :
A.
${\left( {3x + 1} \right)^2} + {\left( {3y} \right)^2} = {a^2} - {b^2}$
B.
${\left( {3x - 1} \right)^2} + {\left( {3y} \right)^2} = {a^2} - {b^2}$
C.
${\left( {3x - 1} \right)^2} + {\left( {3y} \right)^2} = {a^2} + {b^2}$
D.
${\left( {3x + 1} \right)^2} + {\left( {3y} \right)^2} = {a^2} + {b^2}$
2003 JEE Mains MCQ
AIEEE 2003
If the equation of the locus of a point equidistant from the point $\left( {{a_{1,}}{b_1}} \right)$ and $\left( {{a_{2,}}{b_2}} \right)$ is
$\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0$ , then the value of $'c'$ is :
A.
$\sqrt {{a_1}^2 + {b_1}^2 - {a_2}^2 - {b_2}^2} $
B.
${1 \over 2}\left( {{a_2}^2 + {b_2}^2 - {a_1}^2 - {b_1}^2} \right)$
C.
${{a_1}^2 - {a_2}^2 + {b_1}^2 - {b_2}^2}$
D.
${1 \over 2}\left( {{a_1}^2 + {a_2}^2 + {b_1}^2 + {b_2}^2} \right)$.
2003 JEE Mains MCQ
AIEEE 2003
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 JEE Mains MCQ
AIEEE 2003
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 JEE Mains MCQ
AIEEE 2003
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.
$y\left( {\cos \alpha + \sin \alpha } \right) + x\left( {\sin \alpha + \cos \alpha } \right) = a$
2002 JEE Mains MCQ
AIEEE 2002
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 JEE Mains MCQ
AIEEE 2002
The pair of lines represented by $$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$

are perpendicular to each other for :
A.
two values of $a$
B.
$\forall \,a$
C.
for one value of $a$
D.
for no values of $a$
2002 JEE Mains MCQ
AIEEE 2002
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.
${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}$