If the perpendicular drawn from the point $(2, -3)$ to the straight line $4x - 3y + 8 = 0$ meets it at $M(a, b)$ and $a^3 - b^3 = k^3$,then $k=$

Two straight lines are drawn through the point $(0, 2)$ such that the lengths of the perpendiculars from the point $(4, 4)$ to these lines are each equal to $2$ units. The equation of the line joining the feet of these perpendiculars 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:

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 the family of straight lines $ax + by + c = 0$,where $2a + 3b = 4c$,is concurrent at the point $P(l, m)$,then the foot of the perpendicular drawn from $P$ to the line $x + y + 1 = 0$ is

$y-x=0$ 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