The coordinates of the point on the line $x+y+3=0$,whose distance from the line $x+2y+2=0$ is $\sqrt{5}$ units,are

If $p_{1}$ and $p_{2}$ are the lengths of perpendiculars from the origin to the lines $x \sin \theta + y \cos \theta = 5 \cos 2 \theta$ and $x \operatorname{cosec} \theta + y \sec \theta - 5 = 0$ respectively,then $p_{1}^{2} + 4 p_{2}^{2} = $

The distance between the lines $3x + 4y = 9$ and $6x + 8y = 15$ is

If $p$ and $q$ are the lengths of the perpendiculars from the origin to the lines $x \cos \theta - y \sin \theta = k \cos 2 \theta$ and $x \sec \theta + y \csc \theta = k$ respectively,prove that $p^{2} + 4q^{2} = k^{2}$.

Let $R$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$,for which the points $(a^2, a+1)$ lie in $R$,is :

Find the distance between the parallel lines $l(x + y) + p = 0$ and $l(x + y) - r = 0$.