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

The foot of the perpendicular drawn from the origin to the line $3x + y = \lambda \, (\lambda \ne 0)$ is $P$. If the line meets the $x$-axis at $A$ and the $y$-axis at $B$,then the ratio $BP : PA$ is

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:

Find the image of the point $(-1, 3)$ with respect to the line $x - y + 1 = 0$.

The image of a point $A(3, 8)$ in the line $x + 3y - 7 = 0$ 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: