Straight Lines and Pair of Straight Lines

The area of the triangle formed by the line $L$ with the coordinate axes is 12 sq. units. If $L$ passes through the point $(12,4)$ and the product $P$ of $X$ - intercept of $L$ and square of the $Y$-intercept of $L$ is negative, then $P=$

The lines $x+y+4=0, x-2 y-4=0$ and $3 x+4 y-2=0$

The area of the quadrilateral formed by the lines $x+2 y+3=0,2 x+4 y+9=0, x-2 y+3=0$ and $3 x-6 y+11=0$

$(a, b)$ are the new coordinates of the point $(2,3)$ after shifting the origin to the point $(3,2)$ by translation of axes. If $(c, d)$ are the new coordinates of the point $(a, b)$ after rotating the axes through an angle $\frac{\pi}{4}$ ...

If $(-1,-1)$ is the point of intersection of the pair of lines $2 x^2+5 x y-3 y^2+2 g x+2 f y+c=0$. Then $g+f$

Two non-parallel sides of a rhombus are parallel to the lines $x+y-1=0$ and $7 x-y-5=0$. If $(1,3)$ is the centre of the rhombus and one of its vertices $A(\alpha, \beta)$ lies on $15 x-5 y=6$, then one of the possible values of $(\alpha+\b...

$A(2,0), B(0,2), C(-2,0)$ are three points. Let $a, b, c$ be the perpendicular distances from a variable point $P$ on to the lines $A B, B C$ and $C A$ respectively. If $a, b, c$ are in arithmetic progression, then the locus of $P$ is

Two families of lines are given by $a x+b y+c=0$ and $4 a^2+9 b^2-c^2-12 a b=0$. Then, the line common to both the families is

If the equations $3 x^2+2 h x y-3 y^2=0$ and $3 x^2+2 h x y-3 y^2+2 x-4 y+c=0$ represent the four sides of a square, then $\frac{h}{c}=$

A straight line through the point $P(1,2)$ makes an angle $\theta$ with positive X -axis in anticlockwise direction and meets the line $x+\sqrt{3 y}-2 \sqrt{3}=0$ at $Q$. If $P Q=\frac{1}{2}$, then $\theta=$

The lines $x-2 y+1=0,2 x-3 y-1=0$ and $3 x-y+k=0$ are concurrent. The angle between the lines $3 x-y+k=0$ and $m x-3 y+6=0$ is $45^{\circ}$. If $m$ is an integer, then $m-k=$

The line $L \equiv 6 x+3 y+k=0$ divides the line segment joining the points $(3,5)$ and $(4,6)$ in the ratio $-5: 4$. If the point of intersection of the lines $L=0$ and $x-y+1=0$ is $P(g, h)$, then $h=$

If $\tan ^{-1}(2 \sqrt{10})$ is the angle between the lines $a x^2+4 x y-2 y^2=0$ and $a \in Z$, then the product of the slopes of given lines is

If $2 x^2+x y-6 y^2+k=0$ is the transformed equation of $2 x^2+x y-6 y^2-13 x+9 y+15=0$ when the origin is shifted to the point $(a, b)$ by translation of axes, then $k=$

A straight line passing through a point $(3,2)$ cuts $X$ and $Y$ axes at the points $A$ and $B$ respectively. If a point $P$ divides $A B$ in the ratio $2: 3$, then the equation of the locus of point $P$ is

If a line $L$ passing through the point $A(-2,4)$ makes an angel of $60^{\circ}$ with the positive direction of $X$ - axis in anti-clockwise direction and $B(p, q)$ lying in the 3rd quadrant is a point on $L$ at the distance of 6 units from...

If the slopes of the lines represented by the equation $6 x^2+2 h x y+4 y^2=0$ are in the ratio $2: 3$, then the value of $h$ such that both the lines make acute angles with the positive $X$-axis measured in positive direction is

Let $Q$ be the image of a point $P(1,2)$ with respect to the line $x+y+1=0$ and $R$ be the image of $Q$ with respect to the line $x-y-1=0$. If $M$ and $N$ are the mid-points of $P Q$ and $Q R$ respectively, then $M N=$

If the perpendicular drawn from the point $(2,-3)$ to the straight line $4 x-3 y+8=0$ meets it at $M(a, b)$ and $a^3-b^3=k^3$, then $k=$

By shifting the origin to the point $(-1,2)$ through translation of axes, if $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ is the transformed equation of $2 x^2-x y+y^2-3 x+4 y-5=0$, then $2(f+g+h)=$

If $4 x^2+12 x y+9 y^2+2 g x+2 f y-1=0$ represent a pair of parallel lines, then

A variable straight-line $L$ with negative slope passes through the point $(4,9)$ and cuts the positive coordinate axes in $A$ and $B$. If $O$ is the origin, then the minimum value of $O A+O B$ is

If a line $L$ passing through a point $A(2,3)$ intersects another line $4 x-3 y-19=0$ at the point $B$ such that $A B=4$, then the angle made by the line $L$ with positive $X$-axis in anti-clockwise direction is

A line $L$ perpendicular to the line $5 x-12 y+6=0$ makes positive intercept on the $Y$-axis. If the distance from the origin to the line $L$ is 2 units and the angle made by the perpendicular drawn from the origin to the line $L$ with posi...

If $(h, k)$ is the new origin to be chosen to eliminate first degree terms from the equation $S \equiv 2 x^2-x y-y^2-3 x+3 y=0$ by translation and if $\theta$ is the angle with which the axes are to be rotated about the origin in anti-clock...

If the points $A(2,3), B(3,2)$ form a triangle with a variable point $p\left(t, t^2\right)$, where $t$ is a parameter, then the equation of the locus of the centroid of $\triangle A B C$ is

Every point on the curve $3 x+2 y-3 x y=0$ is the centroid of a triangle formed by the coordinate axes and a line $(L)$ intersecting both the coordinates axes. Then, all such lines $(L)$

$O(0,0), B(-3,-1)$ and $C(-1,-3)$ are vertices of a $\triangle O B C$. $D$ is a point on $O C$ and $E$ is a point on $O B$. If the equation of $D E$ is $2 x+2 y+\sqrt{2}=0$, then the ratio in which the line $D E$ divides the altitude of the...

The point $P(\alpha, \beta)(\alpha>0, \beta>0)$ undergoes the following transformations successively.
(a) Translation to a distance of 3 units in positive direction of $X$-axis.
(b) Reflection about the line $y=-x$.
(c) Rotation of axes thr...

If the line passing through the point $(4,-3)$ and having negative slope makes an angle of $45^{\circ}$ with the line joining the points $(1,1),(2,3)$, then the sum of intercepts of that line is

The value of ' $a$ ' for which the equation $\left(a^2-3\right) x^2+16 x y -2 a y^2+4 x-8 y-2=0$ represents a pair of perpendicular lines is

When the origin is shifted to the point $(2, b)$ by translation of axes, the coordinates of the point $(a, 4)$ have changed to $(6,8)$. When the origin is shifted to $(a, b)$ by translation of axes, if the transformed equation of $x^2+4 x y...

If $(4,3)$ and $(1,-2)$ are the end points of a diagonal of a square, then the equation of one of its sides is

$(a, b)$ is the point of concurrency of the lines $x-3 y+3=0, k x+y+k=0$ and $2 x+y-8=0$. If the perpendicular distance from the origin to the line $L=a x-b y+2 k=0$ is $p$, then the perpendicular distance from the point $(2,3)$ to $L=0$ is

The slope of a line $L$ passing through the point $(-2,-3)$ is not defined. If the angle between the lines $L$ and $a x-2 y+3=0(a>0)$ is $45^{\circ}$, then the angle made by the line $x+a y-4 \doteq 0$ with positive $X$-axis in the anti-clo...

Area of the triangle bounded by the lines given by the equations $12 x^2-20 x y+7 y^2=0$ and $x+y-1=0$ is

$L_1 \equiv 2 x+y-3=0$ and $L_2 \equiv a x+b y+c=0$ are two equal sides of an isosceles triangle. If $L_3 \equiv x+2 y+1=0$ is the third side of this triangle and $(5,1)$ is a point on $L_2=$ 0 , then $\frac{b^2}{|a c|}=$

$(-2,-1),(2,5)$ are two vertices of a triangle and $\left(2, \frac{5}{3}\right)$ is its orthocenter. If $(m, n)$ is the third vertex of that triangle, then $m+n$ is equal to.

The slope of one of the pair of lines $2 x^2+h x y+6 y^2=0$ is thrice the slope of the other line, then $h=$

If the ratio of the distances of a variable point $P$ from the point $(1,1)$ and the line $x-y+2=0$ is $1: \sqrt{2}$, then the equation of the locus of $P$ is

$L \equiv x \cos \alpha+y \sin \alpha-p=0$ represents a line perpendicular to the line $x+y+1=0$. If $p$ is positive, $\alpha$ lies in the fourth quadrant and perpendicular distance from $(\sqrt{2}, \sqrt{2})$ to the line, $L=0$ is 5 units,...

If the origin is shifted to the point $\left(\frac{3}{2},-2\right)$ by the translation of axes, then the transformed equation of $2 x^2+4 x y+y^2+2 x-2 y+1=0$ is

$A(1,-2), B(-2,3), C(-1,-3)$ are the vertices of a $\triangle A B C . L_{1}$ is the perpendicular drawn from $A$ to $B C$ and $L_{2}$ is the perpendicular bisector of $A B$. If $(l, m)$ is the point of intersection of $L_{1}$ and $L_{2}$, t...

The area of the parallelogram formed by the lines $L_{1} \equiv \lambda x+4 y+2=0, L_{2} \equiv 3 x+4 y-3=0$, $L_{3} \equiv 2 x+\mu y+6=0, L_{4} \equiv 2 x+y+3=0$, where $L_{1}$ is parallel to $L_{2}$ and $L_{3}$ is parallel to $L_{4}$ is

If the angle between the pair of lines given by the equation $a x^{2}+4 x y+2 y^{2}=0$ is $45^{\circ}$, then the possible values of $a$

$(a, b)$ is the point to which the origin has to be shifted by translation of axes so as to remove the first-degree terms from the equation $2 x^{2}-3 x y+4 y^{2}+5 y-6=0$. If the angle by which the axes are to be rotated in positive direct...

$\beta$ is the angle made by the perpendicular drawn from origin to the line $L \equiv x+y-2=0$ with the positive $X$-axis in the anticlockwise direction. If $a$ is the $X$-intercept of the line $L=0$ and $p$ is the perpendicular distance f...

If $(2,-1)$ is the point of intersection of the pair of lines $2 x^2+a x y+3 y^2+b x+c y-3=0$, then $3 a+2 b+c=$

The line $2 x+y-3=0$ divides the line segment joining the points $A(1,2)$ and $B(-2,1)$ in the ratio $a: b$ at the point $C$. If the point $C$ divides the line segment joining the points $P\left(\frac{b}{3 a},-3\right)$ and $Q\left(-3,-\fra...

If the distance from a variable point $P$ to the point $(4,3)$ is equal to the perpendicular distance from $P$ to the line $x+2 y-1=0$, then the equation of the locus of the point $P$ is