The perpendicular from the origin to the line $y=mx+c$ meets it at the point $(-1, 2)$. Find the values of $m$ and $c$.

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 $a$ and $b$ be any two numbers satisfying $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{4}$. Then,the foot of the perpendicular from the origin on the variable line $\frac{x}{a} + \frac{y}{b} = 1$ lies on

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:

Find the coordinates of the foot of the perpendicular drawn from the point $(7, 8)$ to the line $2x + 3y - 4 = 0$.

Suppose $P(x,y)$ lies on $\sqrt{3} x-y+2=0$ or $\sqrt{3} x+y-2=0$ and is at a distance of $5$ units from their point of intersection. Then the distance from $(0,0)$ to the foot of the perpendicular of $P$ onto the $y$-axis is