Straight Lines — JEE Main PYQs

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ passing through the origin. The tangent of an angle between the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ is :

Let ABC be an isosceles triangle in which A is at $(-1,0),\angle \mathrm{A}=\frac{2\pi }{3},\mathrm{AB}=\mathrm{AC}$ and B is on the positive x -axis. If $\mathrm{BC}=4\sqrt{3}$ and the line BC intersects the line $\mathrm{y}=\mathrm{x}+3$ at $(\alpha ,\beta )$, then …

Let $\alpha ,\beta ,\gamma ,\delta \in \mathrm{Z}$ and let $\mathrm{A}(\alpha ,\beta ),\mathrm{B}(1,0),\mathrm{C}(\gamma ,\delta )$ and $\mathrm{D}(1,2)$ be the vertices of a parallelogram ABCD . If $\mathrm{AB}=\sqrt{10}$ and the points A and C lie on the line …

In a $△ABC$, 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 points A and B are respectively $(4,6)$ and $(\alpha ,\beta )$, then $\alpha +2\beta$ is equal to

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 $\mathrm{g}+\mathrm{c}+\mathrm{h}-\mathrm{f}$ equals

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

The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line L : $9\mathrm{x}+5\mathrm{y}=45$ between the axes. If ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ are the slopes of the lines $l_1$ and $l_2$,then the point of in …

Let the equations of two adjacent sides of a parallelogram ABCD be $2\mathrm{x}-3\mathrm{y}=-23$ and $5\mathrm{x}+4\mathrm{y}=23$. If the equation of its one diagonal AC is $3\mathrm{x}+7\mathrm{y}=23$ and the distance of A from the other diagonal is d , then …

If the point $\left(\alpha ,\frac{7\sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines x $\cos \theta +\mathrm{y}\sin \theta =7,\theta \in \left(0,\frac{\pi }{2}\right)$ between the coordinates axes, then $\alpha$ is equal to

Let $(\alpha ,\beta )$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then $\alpha +2\beta$ and $2\alpha -\beta$ are the roots of the equation

If $(\alpha ,\beta )$ is the orthocentre of the triangle ABC with vertices $\mathrm{A}(3,-7),\mathrm{B}(-1,2)$ and $\mathrm{C}(4,5)$, then $9\alpha -6\beta +60$ is equal to:

If the line ${\ell }_1:3\mathrm{y}-2\mathrm{x}=3$ 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

Let R be a rectangle given by the lines $x=0,x=2$, $y=0$ and $y=5$. Let $A(\alpha ,0)$ and $B(0,\beta ),\alpha \in [0,2]$ and $\beta \in [0,5]$, be such that the line segment AB divides the area of the rectangle R in the ratio 4:1. Then, the mid-point of AB lies on a

A light ray emits from the origin making an angle $30^{\circ }$ with the positive x -axis. After getting reflected by the line $\mathrm{x}+\mathrm{y}=1$, if this ray intersects x -axis at Q , then the abscissa of Q is

Let B and C be the two points on the line $\mathrm{y}+\mathrm{x}=0$ such that B and C are symmetric with respect to the origin. Suppose A is a point on $\mathrm{y}-2\mathrm{x}=2$ such that $△\mathrm{ABC}$ is an equilateral triangle. Then, the area of the …

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

The combined equation of the two lines $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ and ${\mathrm{a}}^{\prime }\mathrm{x}+{\mathrm{b}}^{\prime }\mathrm{y}+{\mathrm{c}}^{\prime }=0$ can be written as …

Let ${\mathrm{m}}_1,{\mathrm{m}}_2$ be the slopes of two adjacent sides of a square of side a such that ${\mathrm{a}}^2+11\mathrm{a}+3\left({\mathrm{m}}_1^2+{\mathrm{m}}_2^2\right)=220$. If one vertex of the square is …

For $\mathrm{t}\in (0,2\pi )$, if ABC is an equilateral triangle with vertices $\mathrm{A}(\sin t,-\cos t),\mathrm{B}(\cos t,\sin t)$ and $\mathrm{C}(\mathrm{a},\mathrm{b})$ such that its orthocentre lies on a circle with centre $\left(1,\frac{1}{3}\right)$, then …

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 line, with the slope greater than one, passes through the point $\mathrm{A}(4,3)$ and intersects the line $\mathrm{x}-y-2=0$ at the point $B$. If the length of the line segment AB is $\frac{\sqrt{29}}{3}$, then B also lies on the line :

The equations of the sides $\mathrm{AB},\mathrm{BC}$ and CA of a triangle ABC are $2\mathrm{x}+\mathrm{y}=0,x+\mathrm{py}=39$ and $\mathrm{x}-\mathrm{y}=3$ respectively and $\mathrm{P}(2,3)$ is its circumcentre. Then which of the following is NOT true :

Let the point $\mathrm{P}(\alpha ,\beta )$ be at a unit distance from each of the two lines $L_1:3x-4y+12=0$, and ${\mathrm{L}}_2:8\mathrm{x}+6\mathrm{y}+11=0$. If P lies below ${\mathrm{L}}_1$ and above $L_2$, then $100(\alpha +\beta )$ is equal to

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 ______.

Let a triangle be bounded by the lines ${\mathrm{L}}_1:2\mathrm{x}+5\mathrm{y}=10$; ${\mathrm{L}}_2:-4\mathrm{x}+3\mathrm{y}=12$ and the line ${\mathrm{L}}_3$, which passes through the point $\mathrm{P}(2,3)$, intersect ${\mathrm{L}}_2$ at A and ${\mathrm{L}}_1$ …

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 …

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:

The distance between the two points A and ${\mathrm{A}}^{\prime }$ which lie on $\mathrm{y}=2$ such that both the line segments AB and ${\mathrm{A}}^{\prime }\mathrm{B}$ (where B is the point (2, 3)) subtend angle $\frac{\pi }{4}$ at the origin, is equal to :

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

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 $△ABC$, then $15(\alpha +\beta )$ is equal to :

Let the circumcentre of a triangle with vertices $\mathrm{A}(\mathrm{a},3),\mathrm{B}(\mathrm{b},5)$ and $\mathrm{C}(\mathrm{a},\mathrm{b}),\mathrm{ab}>0$ be $\mathrm{P}(1,1)$. If the line AP intersects the line BC at the point …

Let $\mathrm{A}\left(\frac{3}{\sqrt{\mathrm{a}}},\sqrt{\mathrm{a}}\right)\mathrm{a}>0$, 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 $\mathrm{D}(3\cos \theta ,a\sin \theta )$ is a point in the four …

Let the area of the triangle with vertices $A(1,\alpha )$, $B(\alpha ,0)$ and $C(0,\alpha )$ be 4 sq. units. If the points $(\alpha ,-\alpha )$, $(-\alpha ,\alpha )$ and $({\alpha }^2,\beta )$ are collinear, then $\beta$ is equal to

Let ABC be a triangle with $\mathrm{A}(-3,1)$ and $\angle \mathrm{ACB}=\theta ,0<\theta <\frac{\pi }{2}$. If the equation of the median through $B$ is $2x+y-3=0$ and the equation of angle bisector of $C$ is $7x-4y-1=0$, then $\tan \theta$ is equal to:

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of the 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 t …

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 $\mathrm{PC}=\sqrt{5}$ inches and $\angle PCB={\tan }^{-1}(2)$.
The acute angle through which the pencil must be rotated about C so that the perpend …

Let a point $P$ be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0)$. If the locus of the point $P$ is a circle of radius $r$, then $4r^2$ is equal to ____

Consider a triangle having vertices $\mathrm{A}(-2,3),\mathrm{B}(1,9)$ and $\mathrm{C}(3,8)$. 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 …

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

The image of the point $(3,5)$ in the line $x-y+1=0$, lies on :

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

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 :

Let $A(1,4)$ and $B(1,-5)$ be two points. Let $P$ be a point on the circle $(x-1{)}^2+(y-1{)}^2=1$ such that $(PA{)}^2+(PB{)}^2$ have maximum value, then the points $P,A$ and $B$ lie on :

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 :

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines
$\mathrm{x}\mathrm{cosec}\alpha -\mathrm{y}\sec \alpha =\mathrm{k}\cot 2\alpha$ and $\mathrm{x}\sin \alpha +\mathrm{y}\cos \alpha =\mathrm{k}\sin 2\alpha$ respectively, then …

In a triangle $PQR$, the co-ordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,-2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x-y+2=0$, then the centre of the circumcircle of the $△PQR$ is:

Let the equation of the pair of lines, $y=px$ and $y=qx$, can be written as $(y-px)(y-qx)=0$. Then the equation of the pair of the angle bisectors of the lines $x^2-4xy-5y^2=0$ is :

The point $\mathrm{P}(\mathrm{a},\mathrm{b})$ undergoes the following three transformations successively :
(a) reflection about the line $\mathrm{y}=\mathrm{x}$
(b) translation through 2 units along the positive direction of $x$-axis.
(c) rotation through angle $\frac{\pi }{4}$ a …

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 :

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 ______.