The perpendicular from the origin to a line meets it at the point $(-2, 9)$. Find the equation of the line.

Find the equation of a line which passes through $(2 \cos^3 \theta, 2 \sin^3 \theta)$ and is perpendicular to the line $x \cos \theta - y \sin \theta = 2 \cos 2 \theta$.

Find the coordinates of the foot of the perpendicular drawn from the point $(0, 5)$ to the line $3x - 4y - 5 = 0$.

If $2x + 3y = 5$ is the perpendicular bisector of the line segment joining the points $A\left(1, \frac{1}{3}\right)$ and $B$,then $B$ is equal to

In a triangle $ABC$,$A(3, 5)$ is a vertex. If the internal angle bisector of $B$ is $y = x$,then which of the following points must lie on the line $BC$?

Let the mirror image of point $A(1, 4)$ in the line $y = x$ be point $B$; the mirror image of point $B$ in the line $y = -x$ be $C$; and the mirror image of $C$ in the $x$-axis be $D$. Then,the area of triangle $ABD$ is ............... $sq. \, units$.