The coordinates of the image point $Q$ of the point $P(-5, 13)$ with respect to the line $2x - 3y - 3 = 0$ are:

If $A(2, -1)$ and $B(6, 5)$ are two points,the ratio in which the foot of the perpendicular from $P(4, 1)$ to $AB$ divides $AB$ is:

The image of the line $x+y-2=0$ in the $y$-axis is

Let $A \left(\frac{3}{\sqrt{a}}, \sqrt{a}\right)$ with $a > 0$ be a fixed point in the $xy$-plane. The image of $A$ in the $y$-axis is $B$,and the image of $B$ in the $x$-axis is $C$. If $D(3 \cos \theta, a \sin \theta)$ is a point in the fourth quadrant such that the maximum area of $\triangle ACD$ is $12$ square units,then $a$ is equal to:

The coordinates of the foot of the perpendicular from the point $(1, 2)$ on the line $x - 3y + 7 = 0$ are

The equation of the straight line which passes through the point $(a \cos^{3} \theta, a \sin^{3} \theta)$ and is perpendicular to $x \sec \theta + y \operatorname{cosec} \theta = a$ is