Straight lines practice · Medium

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

Coordinates of the foot of the perpendicular drawn from (0, 0) to the line joining $(a\cos \mathrm{\alpha },a\sin \mathrm{\alpha }$) and ($a\cos \mathrm{\beta },a\sin \mathrm{\beta }$) are

Family of lines $\lambda x+3 y-6=0(\lambda$ is a real parameter $)$ intersect the lines $x-2 y+3=0$ and $x-y+1=0$ in $P$ and $Q$, then locus of the middle point of $P Q$ is

The straight line $y=m x+c$ cuts the circle $x^2+y^2=a^2$ at the real points if

If sum of distances of a point from the origin and lines $x=2$ is 4 , then its locus is

If $O$ is the origin and $P\left( {{x_1},{y_1}} \right),Q\left( {{x_2},{y_2}} \right)$ are two points, then $OP.OQ\sin \angle POQ =$

The number of integral points (i.e., the points whose $x$ and $y$-coordinates are both integers) within the area (not including the boundary) enclosed by the right branch of hyperbola $x^2-y^2=1$ and the line $x=100$ is

If points $A\left( {{x_1},{y_1}} \right),\;B\left( {{x_2},{y_2}} \right)$ and $C\left( {{x_3},{y_3}} \right)$ are such that $C\left( {{x_3},{y_3}} \right)$ and ${y_1},{y_2},{y_3}$ are in AP, then

Ratio in which the join of $(2,1)$ and $(-1,2)$ is divided by the line $x+3 y+5=0$ is -

The equation of the bisector of that angle between the lines $x+2 y-11=0$, $3 x-6 y-5=0$ which contains the point $(1,-3)$ is

The equation of line through $(1,2)$ and parallel to $3 x-y-4=0$ is-

A line passing through the point $A(9,0)$ makes 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 :

The angle between the lines represented by the equation $4 x^2-24 x y+11 y^2=0$ are

The equation of the line through $(2,3)$, so that the segment of the line intercepted between the axes is bisected at this point, is

The slopes of the lines joining the origin to the points of trisection of the portion of the line $3x + y = 12$ intercepted between the axis are the roots of the equation

The value of $y$, so that the line through $(3, y)$ and $(2,7)$ is parallel to the line through $(-1,4)$ and $(0,6)$, is

The area of triangle formed by the lines $Y=a x$, $x+y-a=0$ and the $Y$-axis is equal to

If $\theta$ denotes the acute angle between the curves, $y = 10 - {x^2}\,\mathrm{ and}\,\mathrm{ }y = 2 + {x^2}$ at a point of their intersection, then $\left| {\tan \theta } \right|$ is equal to

If the lines represented by the equation $2 x^2-3 x y+y^2=0$ make angles $\alpha$ and $\beta$ with $x$-axis, then $\cot ^2 \alpha+\cot ^2 \beta=$

If $\cos \theta=\frac{8}{17}$ and $\theta$ lies between the Ist quadrant, then the value of $\cos \left(30^{\circ}+\theta\right)+\cos \left(45^{\circ}-\theta\right)+\cos \left(120^{\circ}-\theta\right)$ is

A line is such that its segments between the straight lines $5 x-y=4$ and $3 x+4 y-4=0$ is bisected at the points $(1,5)$. Its equation is

$A(4,4), B(3,5)$ and $C(-1,-1)$ are the vertices of

The points $(1,1),(-2,7)$ and $(3,-3)$

The equation of line passing through $\left(-1, \frac{\pi}{2}\right)$ and perpendicular to $\sqrt{3} \sin \theta+2 \cos \theta=\frac{4}{r}$ is

If $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ represents a pair of straight lines equidistant from the origin, then

If $a > 0,\;b > 0$ the maximum area of the triangle formed by the points $O\left( {0,\;0} \right),\;A\left( {a\cos \theta ,\;b\sin \theta } \right)$ and $B\left( {a\cos \theta ,\; - b\sin \theta } \right)$ is (in sq unit)

The line $3x + 2y = 24$ meets y-axis at A and x-axis at B. The perpendicular bisector of AB meets the line through (0,-1) parallel to x-axis at C. The area of the triangle ABC is

From a common point on the lines $x y-3 y-4 x+12=0$ mutually perpendicular lines $L_1$ and $L_2$ are drawn such that they intercept triangles of equal area with coordinate axes. If $\theta_1$ and $\theta_2$ are inclination of $L_1$ and $L_2$ with coordinate axes, then $\left|\theta_1+\theta_2\right|$ is equal to

A straight line $L$ with negative slope passes through the point $(8,2)$ and intersects the positive coordinate axes at point $P$ and $Q$. As $L$ varies the absolute minimum value of $O P+O Q$ is (where, $O$ is origin)

The orthocentre of the triangle with vertices $\left( {2,\frac{{\sqrt 3 - 1}}{2}} \right),\left( {\frac{1}{2}, - \frac{1}{2}} \right)$ and $\left( {2, - \frac{1}{2}} \right)$ is