Straight Lines — JEE Main PYQs

Let a point A lie between the parallel lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ such that its distances from ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C l …

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 …

A rectangle is formed by the lines $\mathrm{x}=0$, $\mathrm{y}=0$, $\mathrm{x}=3$ and $\mathrm{y}=4$. 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 $\left(\frac{1}{2},-5\right)$ …

Let $\mathrm{A}(1,2)$ and $\mathrm{C}(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7\mathrm{x}-\mathrm{y}=14$. If $\mathrm{B}(\alpha ,\beta )$ and $\mathrm{D}(\gamma ,\delta )$ are the other two vertices, then …

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

Let the angles made with the positive x-axis by two straight lines drawn from the point $\mathrm{P}(2,3)$ and meeting the line $\mathrm{x}+\mathrm{y}=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be ${\theta }_1$ and ${\theta }_2$. Then the value of …

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 $|\alpha +\sqrt{2}\beta |$ is

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 :

Let the line ${\mathrm{L}}_1:\mathrm{x}+3=0$ intersect the lines ${\mathrm{L}}_2:\mathrm{x}-\mathrm{y}=0$ and ${\mathrm{L}}_3:3\mathrm{x}+\mathrm{y}=0$ at the points A and B , respectively. Let the bisector of the obtuse angle between the lines ${\mathrm{L}}_2$ and …

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:

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 :

From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ }$ with the line $\mathrm{x}+\mathrm{y}=0$. These rays get reflected from the mirror $x+2y=1$. If the equations of the reflected rays are $ax+by=9$ and …

If a straight line drawn through the point of intersection of the lines $4x+3y-1=0$ and $3x+4y-1=0$, meet the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is:

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2y=2$. If the centroid of $△PQR$ is the point $(\alpha ,\beta )$, then $15(\alpha -\beta )$ is equal to :

Let $\mathrm{A}(6,8),\mathrm{B}(10\cos \alpha ,-10\sin \alpha )$ and $\mathrm{C}(-10\sin \alpha ,10\cos \alpha )$, be the vertices of a triangle. If $\mathrm{L}(\mathrm{a},9)$ and $\mathrm{G}(\mathrm{h},\mathrm{k})$ be its orthocenter and centroid respective …

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 $(\mathrm{QR}{)}^2$ is equal to ___ …

Let the area of a $△\mathrm{PQR}$ with vertices $\mathrm{P}(5,4),\mathrm{Q}(-2,4)$ and $\mathrm{R}(\mathrm{a},\mathrm{b})$ be 35 square units. If its orthocenter and centroid are $\mathrm{O}\left(2,\frac{14}{5}\right)$ and $\mathrm{C}(\mathrm{c},\mathrm{d})$ respecti …

A rod of length eight units moves such that its ends A and B always lie on the lines $\mathrm{x}-\mathrm{y}+2=0$ and $\mathrm{y}+2=0$, respectively. If the locus of the point P, that divides the rod AB internally in the ratio $2:1$ is …

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 $\frac{4}{9}$ o …

Let ABC be a triangle formed by the lines $7\mathrm{x}-6\mathrm{y}+3=0,\mathrm{x}+2\mathrm{y}-31=0$ and $9\mathrm{x}-2\mathrm{y}-19=0$, Let the point $(\mathrm{h},\mathrm{k})$ be the image of the centroid of $△\mathrm{ABC}$ in the line …

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 :

If A and B are the points of intersection of the circle $x^2+y^2-8x=0$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ and a point $P$ moves on the line $2x-3y+4=0$, then the centroid of $△PAB$ lies on the line :

Let the points $\left(\frac{11}{2},\alpha \right)$ lie on or inside the triangle with sides $\mathrm{x}+\mathrm{y}=11,\mathrm{x}+2\mathrm{y}=16$ and $2\mathrm{x}+3\mathrm{y}=29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :

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 $2\mathrm{x}-3\mathrm{y}+\lambda =0$ is $\left(\frac{57}{13},\frac{-40}{13}\right)$, then $|\alpha \lambda |$ is equal to :

Let the area of the triangle formed by a straight line $L:x+by+c=0$ with 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:

Let $A(4,-2),B(1,1)$ and $C(9,-3)$ be the vertices of a triangle ABC. 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 ______.

Consider the lines $\mathrm{x}(3\lambda +1)+\mathrm{y}(7\lambda +2)=17\lambda +5$, $\lambda$ being a parameter, all 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 …

A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines
${\mathrm{L}}_1:2\mathrm{x}+\mathrm{y}+6=0$ and ${\mathrm{L}}_2:4\mathrm{x}+2\mathrm{y}-\mathrm{p}=0,\mathrm{p}>0$, at the points A and B , respectively. I …

Let the three sides of a triangle are on the lines $4x-7y+10=0,x+y=5$ and $7x+4y=15$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $\mathrm{x}=0,\mathrm{y}=0$ and $\mathrm{x}+\mathrm{y}=1$ is

If the orthocentre of the triangle formed by the lines $\mathrm{y}=\mathrm{x}+1,\mathrm{y}=4\mathrm{x}-8$ and $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is at $(3,-1)$, then $\mathrm{m}-\mathrm{c}$ is :

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

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 $(\sqrt{3}+1)x+(\sqrt{3}-1)y=0$ and $(\sqrt{3}-1)x-(\sqrt{3}+1)y+8\sqrt{3}=0$. Then …

A line passing through the point $\mathrm{P}(\mathrm{a},0)$ makes an acute angle $\alpha$ with the positive x-axis. Let this line be rotated about the point P through an angle $\frac{\alpha }{2}$ in the clock-wise direction. If in the new position, the slope of the line is …

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 :

Consider a triangle $ABC$ having the vertices $A(1,2),B(\alpha ,\beta )$ and $C(\gamma ,\delta )$ and angles $\angle ABC=\frac{\pi }{6}$ and $\angle BAC=\frac{2\pi }{3}$. If the points $B$ and $C$ lie on the line $y=x+4$, then ${\alpha }^2+{\gamma }^2$ is equal to __ …

The equations of two sides AB and AC of a triangle ABC are $4x+y=14$ and $3x-2y=5$, respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side BC internally in the ratio $2:1$. The equation of the side BC is :

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 ______.

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 :

If the line segment joining the points $(5,2)$ and $(2,a)$ subtends an angle $\frac{\pi }{4}$ at the origin, then the absolute value of the product of all possible values of a is :

Let a ray of light passing through the point $(3,10)$ reflects on the line $2x+y=6$ and the reflected ray passes 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 ______.

A ray of light coming from the point $\mathrm{P}(1,2)$ gets reflected from the point Q on the x-axis and then passes through the point $\mathrm{R}(4,3)$. If the point $\mathrm{S}(\mathrm{h},k)$ is such that PQRS is a parallelogram, then ${\mathrm{hk}}^2$ is equal to :

Two vertices of a triangle ABC are $\mathrm{A}(3,-1)$ and $\mathrm{B}(-2,3)$, and its orthocentre is $\mathrm{P}(1,1)$. If the coordinates of the point C are $(\alpha ,\beta )$ and the centre of the circle circumscribing the triangle PAB is $(\mathrm{h},\mathrm{k})$, then the val …

The distance of the point $(2,3)$ from the line $2x-3\mathrm{y}+28=0$, measured parallel to the line $\sqrt{3}\mathrm{x}-\mathrm{y}+1=0$, is equal to

Let A be the point of intersection of the lines $3\mathrm{x}+2\mathrm{y}=14,5\mathrm{x}-\mathrm{y}=6$ and B be the point of intersection of the lines $4x+3y=8,6x+y=5$. The distance of the point $\mathrm{P}(5,-2)$ from the line AB is

Let $\mathrm{A}(\mathrm{a},\mathrm{b})$, $\mathrm{B}(3,4)$ and $(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $\mathrm{P}(2\mathrm{a}+3,7\mathrm{b}+5)$ from the line $2\mathrm{x}+3\mathrm{y}-4=0$ measured …

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$, meet the positive co-ordinate axes at the point A and B. Then the minimum value of $\mathrm{OA}+\mathrm{OB}$, where O is the origin, is equal to

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 ______.

A line passing through the point $\mathrm{A}(9,0)$ 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

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 :

If the sum of squares of all real values of $\alpha$, for which the lines $2x-y+3=0,6x+3y+1=0$ and $\alpha \mathrm{x}+2\mathrm{y}-2=0$ do not form a triangle is p , then the greatest integer less than or equal to p is ______