The equation of the straight line which is perpendicular to the line $5x - 2y = 7$ and passing through the point of intersection of the lines $2x + 3y - 1 = 0$ and $3x + 4y - 6 = 0$ is

The equation of the straight line in the normal form which is parallel to the lines $x+2y+3=0$ and $x+2y+8=0$ and divides the distance between these two lines in the ratio $1:2$ internally is

$A$ is a point on either of two lines $y + \sqrt{3} |x| = 2$ at a distance of $\frac{4}{\sqrt{3}}$ units from their point of intersection. The coordinates of the foot of the perpendicular from $A$ on the bisector of the angle between them are

Let $A \equiv (0, 1)$,$B \equiv (2, 0)$,and point $P$ be a point on the line $4x + 3y + 9 = 0$. Find the coordinates of point $P$ such that $|PA - PB|$ is maximized.

For a point $P(x, y)$ in the plane,let $d_1(P)$ and $d_2(P)$ be the distances of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2 \leq d_1(P)+d_2(P) \leq 4$ is:

The equation of the line passing through the point of intersection of the lines $3x - 4y + 1 = 0$ and $5x + y - 1 = 0$ and making equal non-zero intercepts on the coordinate axes is