Let $M$ be the foot of the perpendicular drawn from the point $(5, -7)$ to the line $3x - 5y + 1 = 0$. Then the perpendicular distance from $M$ to the line $2x + 5y - 3 = 0$ is

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

Let $\alpha$ and $\beta$ be integers satisfying $0 < \beta < \alpha$. Let $P(\alpha, \beta)$ be a point. Let $Q$ be the reflection of $P$ in the line $y = x$,$R$ be the reflection of $Q$ in the $y$-axis,$S$ be the reflection of $R$ in the $x$-axis,and $T$ be the reflection of $S$ in the $y$-axis. If the area of the convex pentagon $PQRST$ is $187 \ sq. \ units$,then the value of $\alpha + \beta^2$ is:

Let $L$ be the line $y = 2x$ in the two-dimensional plane.
Statement $1$: The image of the point $(0, 1)$ in $L$ is the point $\left( \frac{4}{5}, \frac{3}{5} \right)$.
Statement $2$: The points $(0, 1)$ and $\left( \frac{4}{5}, \frac{3}{5} \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.

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:

$y-x=0$ is the equation of a side of a triangle $ABC$. The orthocentre and circumcentre of the triangle $ABC$ are respectively $(5,8)$ and $(2,3)$. The reflection of the orthocentre with respect to any side of the triangle lies on its circumcircle. Then the radius of the circumcircle of the triangle is