Straight Lines and Pair of Straight Lines

$A$ straight line passing through the origin $O$ meets the parallel lines $4 x+2 y=9$ and $2 x+y+6=0$ at the points $P$ and $Q$ respectively. Then, the point $O$ divides the line segment $P Q$ in the ratio

If a straight line is at a distance of 10 units from the origin and the perpendicular drawn from the origin to it makes an angle $\frac{\pi}{4}$ with the negative $X$-axis in the negative direction, then the equation of that line is

If one of the lines given by the pair of lines $3 x^2-2 y^2+a x y=0$ is making an angle $60^{\circ}$ with $X$-axis, then $a=$

The image of a point $(2,-1)$ with respect to the line $x-y+1=0$ is

The coordinate axes are rotated about the origin in the counter clockwise direction through an angle $60^{\circ}$. If a and $b$ are the intercepts made on the new axes by a straight line whose equation referred to the original axes is $x+y=...

$P Q R$ is a right-angled isosceles triangle with right angle at $P(2,1)$. If the equation of the line $Q R$ is $2 x+y=3$, then the equation representing the pair of lines $P Q$ and $P R$ is

If the equation of the pair of straight lines intersecting at ( $a, b$ ) and perpendicular to the pair of lines $3 x^2-4 x y+5 y^2=0$ is $l x^2+2 n x y+m y^2-32 x-26 y+c=0$, then $\frac{a+b+c}{l+h+m}=$

The points $(2,3)$ and $\left(-4,-\frac{4}{3}\right)$ lie on the opposite sides of the line $L \equiv 5 x-6 y+k=0$ and k is an integer. If the points $(1,2)$ and $(4,5)$ lie on the same side of the line $L=0$, then the perpendicular distanc...

By shifting the origin to the point $(2,3)$ through translation of axes. If the equation or the curve $x^2+3 x y-2 y^2+4 x-y-20=0$ is transformed to the form $A x^2+B x y+C y^2+D x+E y+F=0$, then $D+E+F=$

If the incentre of the triangle formed by the lines $x-2=0, x+y-1=0, x-y+3=0$ is $(\alpha, \beta)$, then $\beta=$

The angle made by a line $L$ with positive $X$-axis measured in the positive direction is $\frac{\pi}{6}$ and the intercept made by $L$ on $Y$-axis is negative. IF $L$ is at a distance of 5 units from the origin, then the perpendicular dist...

If the axes are translated to the orthocentre of the triangle formed by the points $\mathrm{A}(7,5), \mathrm{B}(-5,-7)$ and $C(7,-7)$, then the coordinates of the incentre of the triangle in the new system are

The lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$ respectively. The bisector of the angle between $L_1$ and $L_2$ divides the line segment $P Q$ internally at $R$.
Statement $I P R: R Q=2 \sqrt{2}: \sq...

If $2 x^2+3 x y-2 y^2-5 x+2 f y-3=0$ represents a pair of straight lines, then one of the possible values of $f$ is

$L_1$ and $L_2$ are two lines having slopes 2 and $-\frac{1}{2}$ respectively. If both $L_1$ and $L_2$ are concurrent with the lines $x-y+2=0$ and $2 x+y+3=0$, then sum of the absolute values of the intercepts made by the lines $L_1$ and $L...

$L_1 \equiv a x-3 y+5=0$ and $L_2 \equiv 4 x-6 y+8=0$ are two parallel lines. If $p, q$ are the intercepts made by $L_1=0$ and $m, n$ are the intercepts made by $L_2=0$ on the $X$, $Y$-coordinate axes respectively, then the equation of the ...

If $(h, k)$ is the image of the point $(2,-3)$ with respect to the line $5 x-3 y=2$, then $h+k=$

If the angle between the pair of lines $2 x^2+2 h x y+2 y^2-x+y-1=0$ is $\tan ^{-1}\left(\frac{3}{4}\right)$ and $h$ is a positive rational number, then the point of intersection of these two lines is

If the pair of lines $a x^2-7 x y-3 y^2=0$ and $2 x^2+x y-6 y^2=0$ have exactly one line in common and ' $a$ ' is an integer, then the equation of the pair of bisectors of the angles between the lines $a x^2-7 x y-3 y^2=0$ is

The point $P(4,1)$ undergoes the following transformations in succession :
(i) origin is shifted to the point $(1,6)$ by translation of axes.
(ii) translation through a distance of 2 units along the positive direction of $X$-axis.
(iii) rot...

If $\left(\frac{2}{3}, 0\right)$ is the centroid of the triangle formed by the lines $4 x^2-y^2=0$ and $l x+m y+n=0$, then, $l+m+n=$

The triangle formed by the lines $2 x^2+x y-6 y^2=0$ and $x+y-1=0$ is

The equation of the base of an equilateral triangle is $x+y=2$ and its opposite vertex is $(2,1)$. If $m_1, m_2$ are the slopes of the other two sides and the length of its side is $a$, then $\left|m_1-m_2\right|+a \sqrt{2}=$

Let $A(5,4)$ and $B(5,-4)$ be two points.
If $P$ is a point in the coordinate plane such that $\sqrt{A P B}=\frac{\pi}{4}$, then the point $P$ lies on the curve

If the perpendicular distances from the points $(2,3)$, $(4, a)$ and $(\alpha, \beta)$ on to the line $3 x+4 y-3=0$ are equal and $4 \alpha-3 \beta+1=0$, then sum of all possible values of $a, \alpha$ and $\beta$ is

If the locus of a point which is equidistant from the coordinate axes forms a triangle with the line $y=3$, then the area of the triangle is

If the combined equation of the lines joining the origin to the point of intersection of the curve $x^2+y^2-2 x-4 y+2=0$ and the line $x+y-2=0$ is $\left(l_1 x+m_1 y\right)\left(l_2 x+m_2 y\right)=0$, then $l_1+l_2+m_1+m_2=$

If $\alpha$ is the angle made by the perpendicular drawn from origin to the line $12 x-5 y+13=0$ with the positive $X$-axis in anti-clockwise direction, then $\alpha=$

If the equation of the pair of lines passing through $(1,1)$ and perpendicular to the pair of line $2 x^2+x y-y^2-x+2 y-1=0$ is $a x^2+2 h x y+b y^2+2 g x+3 y=0$, then $\frac{b}{a}=$

$A(-2,3)$ is a point on the line $4 x+3 y-1=0$. If the points on the line that are 10 units away from the point $A$ are ( $x_1, y_1$ ) and ( $x_2, y_2$ ), then $\left(x_1+y_1\right)^2+\left(x_2+y_2\right)^2=$

If $M$ is the foot of the perpendicular drawn from the origin to the line $x-2 y+3=0$ which meets the $X$ and $Y$-axes at $A$ and $B$, respectively, then $A M=$

A straight line passing through a fixed point $(2,3)$ intersects the coordinate axes at points $P$ and $Q$. If $O$ is the origin and $R$ is a variable point such that $O P R Q$ is a rectangle, then the locus of $R$ is

If the lines $x+2 a y+a=0, x+3 b y+b=0$, $x+4 c y+c=0$ are concurrent, then $a, b, c$ are in

If the angle between the lines joining the origin to the points of intersection of $x+2 y+\lambda=0$ and $2 x^2-2 x y+3 y^2+2 x-y-1=0$ is $\frac{\pi}{2}$, then a value of $\lambda$. is

One line of the pair of lines $x^2+x y-2 y^2=0$ is perpendicular to one line of the pair of lines $3 y^2-5 x y-2 x^2=0$ If the combined equation of the two lines other than those two perpendicular lines is $a x^2+2 h x y+b y^2=0$, then $a+2...

If $a x^2+2 h x y-2 a y^2+3 x+15 y-9=0$ represents a pair of lines intersecting at $(1,1)$, then $a h=$

If two sides of a triangle are represented by $3 x^2-5 x y+2 y^2=0$ and its orthocentre is $(2,1)$, then the equation of the third side is

If the lines $x+y-2=0,3 x-4 y+1=0$ and $5 x+k y-7=0$ are concurrent at $(\alpha, \beta)$, then equation of the line concurrent with the given lines and perpendicular to $k x+y-k=0$ is

A line $L_1$ passing through the point of intersection of the lines $x-2 y+3=0$ and $2 x-y=0$ is parallel to the line $L_2$. If $L_2$ passes through origin and also through the point of intersection of the lines $3 x-y+2=0$ and $x-3 y-2=0$,...

The transformed equation of $3 x^2-4 x y=r^2$ when the coordinate axes are rotated about the origin through an angle of $\tan ^{-1}(2)$ in positive direction is

If $A(1,0), B(0,-2)$ and $C(2,-1)$ are three fixed points, then the equation of the locus of a point $P$ such that area of $\triangle P A B$ is equal to area of $\triangle P A C$ is

If the intercept of a straight line $L$ made between the straight lines $5 x-y-4=0$ and $3 x+4 y-4=0$ is bisected at the point $(1,5)$, then the equation of $L$ is

$A$ line $L$ passes through the point $P(1,2)$ and makes an angle of $60^{\circ}$ with $O X$ in the positive direction. $A$ and $B$ are two points lying on $L$ at a distance of 4 units from $P$. If $O$ is the origin, then the area of $\tria...

If $P$ is a variable point which is at a distance of 2 units. from the line $2 x-3 y+1=0$ and $\sqrt{13}$ units from the point $(5,6)$, then the equation of the locus of $P$ is

The equation $(2 p-3) x^2+2 p x y-y^2=0$ represents a pair of distinct lines

If the equation $3 x^2+4 y^2-x y+k=0$ is the transformed equation of $3 x^2+4 y^2-x y-5 x-7 y+2=0$ after shifting the origin to the point $(\alpha, \beta)$ by the translation of axes, then $\alpha+\beta-k=$

$P(6,4)$ is a point on the line $x-y-2=0$. If $A(\alpha, \beta)$ and $B(\gamma, \delta)$ are two points on this line lying on either side of $P$ at a distance of 4 units from $P$, then $\alpha^2+\beta^2+\gamma^2+\delta^2=$

If the straight line $2 x+3 y+1=0$ bisects the angle between two other straight lines one of which is $3 x+2 y+4=0$, then the equation of the other straight line is

If the slope of both the line given by $x^2+2 h x y+6 y^2=0$ are options and the angle between these lines is $\tan ^{-1}\left(\frac{1}{7}\right)$, then the product of the perpendiculars draw from the point $(1,0)$ to the given pair of line...

If the transformed equation of the equation $2 x^2+3 x y-2 y^2-17 x+6 y+8=0$ after translating the coordinate axes to a new origin ( $\alpha, \beta$ ) is $a X^2+2 h X Y+b Y^2+c=0$, then $3 \alpha+c=$