$A$ straight line through the origin $O$ meets the parallel lines $4x + 2y = 9$ and $2x + y + 6 = 0$ at $P$ and $Q$ respectively. The point $O$ divides the segment $PQ$ in the ratio

The length of the perpendicular from $(1, -2)$ to the line $12x + 5y + 63 = 0$ is

Let the origin be the centroid of an equilateral triangle $ABC$ and one of its sides be along the straight line $x+y=3$. If $R$ and $r$ are its circumradius and inradius respectively,then $R+r=$

Let $d_{1}$ and $d_{2}$ be the lengths of the perpendiculars drawn from any point on the line $7x - 9y + 10 = 0$ to the lines $3x + 4y = 5$ and $12x + 5y = 7$, respectively. Then,

The base of an equilateral triangle is represented by the equation $2x - y - 1 = 0$ and its vertex is $(1, 2)$. Then,the length (in units) of the side of the triangle is:

Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $x + y = 3$. If $R$ and $r$ are the radius of the circumcircle and incircle respectively of $\Delta ABC$,then $(R + r)$ is equal to ..... .