If the coordinates of $A$ and $B$ are $(1, 1)$ and $(5, 7)$,then the equation of the perpendicular bisector of the line segment $AB$ is

The coordinates of the foot of the perpendicular drawn from $(0,0)$ to the line joining $(a \cos \alpha, a \sin \alpha)$ and $(a \cos \beta, a \sin \beta)$ are

Let $ABC$ be a triangle formed by the lines $7x-6y+3=0$,$x+2y-31=0$,and $9x-2y-19=0$. Let the point $(h, k)$ be the image of the centroid of $\Delta ABC$ in the line $3x+6y-53=0$. Then $h^2+k^2+hk$ is equal to

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:

Suppose $A$ and $B$ are two points on the line $2x - y + 3 = 0$ and $P(1, 2)$ is a point such that $PA = PB$. Then,the mid-point of $AB$ is

If $(h, k)$ is the image of the point $(2, -3)$ with respect to the line $5x - 3y = 2$,then $h + k =$