Two sides of a rhombus are along the lines, $x -y+ 1 = 0$ and $7x-y-5 =0.$ If its diagonals intersect at $(-1,-2),$ then which one of the following is a vertex of this rhombus?

Given three points $P, Q, R$ with $P(5, 3)$ and $R$ lies on the $x-$ axis. If equation of $RQ$ is $x -2y = 2$ and $PQ$ is parallel to the $x-$ axis, then the centroid of $\Delta PQR$ lies on the line

If one vertex of an equilateral triangle of side $'a'$ lies at the origin and the other lies on the line $x - \sqrt{3} y = 0$ then the co-ordinates of the third vertex are :

If the middle points of the sides $BC,\, CA$ and $AB$ of the triangle $ABC$ be $(1, 3), \,(5, 7)$ and $(-5, 7)$, then the equation of the side $AB$ is

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4$, is $a x^2+b y^2+c x y+d x+e y+170=0$, then the value of $\mathrm{a}^2+2 \mathrm{~b}+3 \mathrm{c}+4 \mathrm{~d}+\mathrm{e}$ is equal to:

In an isosceles triangle $ABC, \angle C = \angle A$ if point of intersection of bisectors of internal angles $\angle A$ and $\angle C$ divide median of side $AC$ in $3 : 1$ (from vertex $B$ to side $AC$), then value of $cosec \ \frac{B}{2}$ is equal to