Pair of straight lines practice · Medium

If the line L: $\frac{x}{5}+\frac{y}{b}=1$ passes through $(13,32)$ and line $\mathrm{K}: \frac{x}{c}+\frac{y}{3}=1$ is parallel to $\mathrm{L}$, then the distance between $\mathrm{L}$ and $\mathrm{K}=\ldots \ldots .$.

Area of the parallelogram whose sides are $x \cos \alpha+y \sin \alpha=p \,\,x \cos \alpha+y \sin \alpha=q$, $x \cos \beta+y \sin \beta=r$ and $x \cos \beta+y \sin \beta=s$ is

Which of the following pairs of straight lines intersects at right angles?

The slope of a line is 3 times the slope of the other line and the tangent of the angle between them is $4 / 13$, sum of their slopes is equal to

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

One diagonal of a square is the portion of the line $3 x+2 y=12$ intercepted between the axes. The coordinates of the extremity of the other diagonals not lying in the first quadrant are

The angle between the lines joining the origin to the points of intersection of the line $\sqrt{3} x+y=2$ and the curve $y^2-x^2=4$ is

If $\alpha, \beta>0$ and $\alpha<\beta$ and $\alpha x^2+4 \gamma x y+\beta y^2+4 p(x+y+1)=0$ represents a pair of straight lines, then

The equation of the line which bisects the obtuse angle between the lines $x-2 y+4=0$ and $4 x-3 y+2=0$, is

A light ray coming along the line $3 x+4 y=5$, gets reflected from the line $a x+b y=1$ and goes along the line $5 x-12 y=10$, then

If $y=m x$ be one of the bisectors of the angle between the lines $a x^2-2 h x y+b y^2=0$, then

Mixed term $x y$ is to be removed from the general equation $a x^2+b y^2+2 h x y+2 g x+2 f y+c=0$, one should rotate the axes through an angle $\theta$ given by $\tan 2 \theta=$

If $a, x_1, x_2$ are in G.P. with common ratio $r$, and $b, y_1, y_2$ are in G.P. with common ratio $s$ where $s-r=2$, then the area of the triangle with vertices $(a, b)$, $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is

If the system of equations $a x+h y+g=0, h x+b y+f=0$ and $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c+t$ $=0$ has a unique solution, then

The value of $\lambda$ so that $3 x^2+7 x y+\lambda y^2+5 x+5 y+2=0$ represents a pair of straight lines is

Given, a straight lines and a fixed point $O$, a straight line is drawn through $O$ meeting lines in the points $R_1, R_2, R_3, \ldots, R_n$ and on it a point $R$ is taken such that $\frac{n}{O R}=\frac{1}{O R_1}+\frac{1}{O R_2}+\ldots+\frac{1}{O R_n}$, then the locus of $R$ is

The equation $3 x^2+7 x y+2 y^2+5 x+5 y+2=0$ represents

The straight lines $7 x-2 y+10=0$ and $7 x+2 y-10=0$ forms and isosceles triangle with the line $y=2$, area of this triangle is equal to

If the lines $x=k ; k=1,2 \ldots, n$ meet the line $y=3 x+4$ at the points $A_k\left(x_k, y_k\right), k=1,2, \ldots, n$ then the ordinate of the centre of Mean position of the points $A_k$, $k=1,2, \ldots, n$ is

The equation of the lines which are passing through the origin and which are perpendicular to the pair of straight lines $x^2-3 x y+2 y^2=0$ is

The area of the triangle formed by the lines $4 x^2-9 x y-9 y^2=0$ and $x=2$ is equal to

If the pairs of lines $x^2+2 x y+\lambda y^2=0$ and $\lambda x^2+2 x y+y^2=0$ have exactly one line in common, then the joint equation of the other two lines is given by

If the straight lines $a x^2+2 h x y+b y^2$ $+2 g x+2 f y+c=0$ intersect on the $X$-axis, then

Two lines represented by the equation $x^2-y^2-2 x+1=0$ are rotated about the point $(1,0)$, the line making the bigger angle with the positive direction of the $x$-axis being turned by $45^{\circ}$ in the clockwise sense and the other line being turned by $15^{\circ}$ in the anticlockwise sense. The combined equation of the pair of lines in their new position, is

The equation $x^2-3 x y+\lambda y^2+3 x-5 y+2=0$ when $\lambda$ is a real number, represents a pair of straight lines. If $\theta$ is the angle between the lines, then $\operatorname{cosec}^2 \theta=$

Distance between the lines represented by the equation $x^2+2 \sqrt{3} x y+3 y^2-3 x-3 \sqrt{3} y-4=0$ is

If the distance of two lines passing through origin from the point $\left(x_1, y_1\right)$ is '$d$', then the equation of lines is

A line is drawn perpendicular to the line $y=5 x$, meeting the coordinate axes at $A$ and $B$. If the area of $\triangle O A B$ is 10 sq units, where $O$ is the origin, then equation of drawn line is

If the lines joining origin to the points of intersection of the line $f x-g y=\lambda$ and the curve $x^2+h x y-y^2+g x+f y=0$ are mutually perpendicular, then