Straight Lines and Pair of Straight Lines

If one of the lines represented by $a x^2+2 h x y+b y^2=0$ bisects the angle between the positive coordinates axes, then

If the distance of a variable point $P$ from a point $A(2,-2)$ is twice the distance of $P$ from $Y$-axis, then the equation of locus of $P$ is

A pair of lines drawn through the origin forms a right angled isosceles triangle with right angle at the origin with the line $2 x+3 y=6$. The area (in sq units) of the triangle thus formed is

The combined equation of the bisectors of the angles between the lines joining the origin to the points of intersection of the curve $x^2+y^2+x y+x+3 y+1=0$ and the line $x+y+2=0$ is

The equation of the perpendicular bisectors of the sides $A B$ and $A C$ of $\triangle A B C$ are $x-y+5=0$ and $x+2 y=0$ respectively, If the coordinates of $A$ are $(1,-2)$, then the equal of the line $B C$ is

If the straight line passing through $P(3,4)$ makes an angle $\frac{\pi}{6}$ with the positive $X$-axis in anti-clockwise direction and meets the line $12 x+5 y+10=0$ at $Q$, then the length of the segment $P Q$ is

The origin is shifted to the point $(2,3)$ by translation of axes and then the coordinate axes are rotated about the origin through an angle $\theta$ in the counter - clockwise sense. Due to this if the equation $3 x^2+2 x y+3 y^2-18 x-22 y...

The locus of the mid-point of the portion of the line $x \cos \alpha+y \sin \alpha=p$ intercepted by the coordinate axes, where $p$ is a constant, is

The equation of the side of an equilateral triangle is $x+y=2$ and one vertex is $(2,-1)$. The length of the side is

The orthocentre of the triangle formed by lines $x+y+1=0, x-y-1=0$ and $3 x+4 y+5=0$ is

The length of $x$-intercept made by pair of lines $2 x^2+x y-6 y^2-2 x+17 y-12=0$ is

If the slope of one of the pair of lines represented by $2 x^2+3 x y+K y^2=0$ is 2 , then the angle between the pair of lines is

The equation of the locus of points which are equidistant from the point $(2,3)$ and $(4,5)$ is

If the reflection of a point $A(2,3)$ in $X$-axis is $B$, reflection of $B$ in the line $x+y=0$ is $C$ and the reflection of $C$ in $x-y=0$ is $D$, then the point of intersection of the lines $C D, A B$ is

The equation of a line which makes an angle of $45^{\circ}$ with each of the pair of lines $x y-x-y+1=0$ is

The locus of a variable point which forms a triangle of fixed area with two fixed points is

$A$ line $L$ passing through the point $P(-5,-4)$ cuts the lines $x-y-5=0$ and $x+3 y+2=0$ respectively at $Q$ and $R$ such that $\frac{18}{P Q}+\frac{15}{P R}=2$, then slope of line $L$ is

If the slope of one of the lines in the pair of lines $8 x^2+a x y+y^2=0$ is thrice the slope of the second line, then $a$ is equal to

$P$ is a variable point such that the distance of $P$ from $A$ $(4,0)$ is twice the distance of $P$ from $B(-4,0)$. If the line $3 y-3 x-20=0$ intersects the locus of $P$ at the points $C$ and $D$, then the distance between $C$ and $D$ is

When the origin is shifted to $(h, k)$ by translation of axes, the transformed equation of $x^2+2 x+2 y-7=0$ does not contain $x$ term and constant term. Then, $(2 h+k)=$

The area of the triangle formed by the lines represented by $3 x+y+15=0$ and $3 x^2+12 x y-13 y^2=0$ is

Let $\alpha \in R$. If the line $(\alpha+1) x+\alpha y+\alpha=1$ passes through a fixed point $(h, k)$ for all $\alpha$, then $h^2+k^2=$

If all chords of the curve $2 x^2-y^2+3 x+2 y=0$, which subtend a right angle at the origin always passing through the point $(\alpha, \beta)$, then $(\alpha, \beta)=$

If the pair of lines represented by $3 x^2-5 x y+P y^2=0$ and $6 x^2-x y-5 y^2=0$ have one line in common, then the sum of all possible value of $P$ is

For $\lambda, \mu \in R,(x-2 y-1)+\lambda(3 x+2 y-11)=0$ and $(3 x+4 y-11)+\mu(-x+2 y-3)=0$ represent two families of lines. If the equation of the line common to both the families is $a x+b y-5=0$. Then, $2 a+b=$

Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $x y$ form from the equation $3 x^2+2 \sqrt{3} x y+y^2=0$. Then, in the new coordinate system the equation $x^2+y^2+2 x y=2$ is transformed to

$P$ is a point on $x+y+5=0$, whose perpendicular distance from $2 x+3 y+3=0$ is $\sqrt{13}$, then the coordinates of $P$ are

A line $L$ passing through the point $(2,0)$ makes an angle $60^{\circ}$ with the line $2 x-y+3=0$. If $L$ makes an acute angle with the positive X-axis in the anti-clockwise direction, then the $Y$-intercept of the line $L$ is

A line $L$ intersects the lines $3 x-2 y-1=0$ and $x+2 y+1=0$ at the points $A$ and $B$. If the point $(1,2)$ bisects the line segment $A B$ and $\frac{x}{a}+\frac{y}{b}=1$ is the equation of the line $L$, then $a+2 b+1=$

If the equation of the pair of straight lines passing through the point $(1,1)$ and perpendicular to the pair of lines $3 x^2+11 x y-4 y^2=0$ is $a x^2+2 h x y+b y^2+2 g x+2 f y+12=0$, then $2(a-h+b-g+f-12)=$

If the slope of one line 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 origin is shifted to a point $P$ by the translationd axes to remove the $y$-term from the equation $x^2-y^2+2 y-1=0$, then the transformed equation of it is

The circumcentre of the triangle formed by the lines $x+y+2=0,2 x+y+8=0$ and $x-y-2=0$ is

If the origin is shifted to remove the first degree terms from the equation $2 x^2-3 y^2+4 x y+4 x+4 y-14=0$, then with respect to this new coordinate system the transformed equation of $x^2+y^2-3 x y+4 y+3=0$ is

If the area of the triangle formed by the straight lines $-15 x^2+4 x y+4 y^2=0$ and $x=\alpha$ is 200 sq unit, then $|\alpha|=$

If the line $2 x-3 y+5=0$ is the perpendicular bisector of the line segment joining $(1,-2)$ and $(\alpha, \beta)$, then $\alpha+\beta=$

The equation for straight line passing through the point of intersection of the lines represented by $x^2+4 x y+3 y^2-4 x-10 y+3=0$ and the point $(2,2)$ is

A ray of light passing through the point $(2,3)$ reflects on $Y$-axis at a point $P$. If the reflected ray passes through the point $(3,2)$ and $P=(a, b)$, then $5 b=$

The point $(a, b)$ is the foot of the perpendicular drawn from the point $(3,1)$ to the line $x+3 y+4=0$. If $(p, q)$ is the image of $(a, b)$ with respect to the line $3 x-4 y+11=0$, then $\frac{p}{a}+\frac{q}{b}=$

The area (in sq units) of the triangle formed by the lines $6 x^2+13 x y+6 y^2=0$ and $x+2 y+3=0$ is

Point $(-1,2)$ is changed to $(a, b)$, when the origin is shifted to the point $(2,-1)$ by translation of axes, Point $(a, b)$ is changed to $(c, d)$, when the axes are rotated through an angle of $45^{\circ}$ about the new origin, $(c, d)$...

If a variable straight line passing through the point of intersection of the lines $x-2 y+3=0$ and $2 x-y-1=0$ intersects the $X, Y$-axes at $A$ and $B$ respectively, then the equation of the locus of a point which divides the segment $A B$...

The area of the triangle formed by the pair of lines $23 x^2-48 x y+3 y^2=0$ with the line $2 x+3 y+5=0$, is

$2 x^2-3 x y-2 y^2=0$ represents two lines $L_1$ and $L_2$. $2 x^2-3 x y-2 y^2-x+7 y-3=0$ represents another two lines $L_3$ and $L_4$. Let $A$ be the point of intersection of lines $L_1, L_3$ and $B$ be the point of intersection of lines $...

$A$ is the point of intersection of the lines $3 x+y-4=0$ and $x-y=0$. If a line having negative slope makes an angle of $45^{\circ}$ with the line $x-3 y+5=0$ and passes through $A$, then its equation is

If the lines $3 x+y-4=0, x-\alpha y+10=0, \beta x+2 y+4=0$ and $3 x+y+k=0$ represent the sides of a square, then $\alpha \beta(k+4)^2=$

The equal sides of an isosceles triangle are given by equations $7 x-y+3=0$ and $x+y-3=0$. If the slope $m$ of the third side is an integer, then $m=$

The combined equation of the lines passing through the point $(3,4)$ and each making an angle $45^{\circ}$ with the line $x+y+1=0$ is-

In $\triangle A B C$ the coordinates of the vertex $A$ are $(-3,1)$. If the equation of the median through $B$ is $2 x+y-3=0$ and the equation of the bisector of $\angle C$ is $7 x-4 y-1=0$, then the equation of the side $B C$ is

If the lines given by $\left(x^2+y^2\right) \sin ^2 \alpha=(x \cos \alpha-y \sin \alpha)^2$ are perpendicular to each other, then $\sin ^2 \alpha+\tan ^2 \alpha=$