The area of the triangle formed by the line $x\sin \alpha + y\cos \alpha = \sin 2\alpha $and the coordinates axes is

The base $BC$ of a triangle $ABC$ is bisected at the point $(p, q)$ and the equation to the side $AB \,\,ane\,\, AC$ are $px + qy = 1 \,\,ane\,\, qx + py = 1$ . The equation of the median through $A$ is :

In a rectangle $A B C D$, the coordinates of $A$ and $B$ are $(1,2)$ and $(3,6)$ respectively and some diameter of the circumscribing circle of $A B C D$ has equation $2 x-y+4=0$. Then, the area of the rectangle is

Coordinates of the vertices of a quadrilateral are $(2, -1), (0, 2), (2, 3)$ and $(4, 0)$. The angle between its diagonals will be

Let the equations of two adjacent sides of a parallelogram $A B C D$ be $2 x-3 y=-23$ and $5 x+4 y$ $=23$. If the equation of its one diagonal $AC$ is $3 x +$ $7 y=23$ and the distance of A from the other diagonal is $d$, then $50 d ^2$ is equal to $........$.

The base of an equilateral triangle with side $2 a$ lies along the $y$ -axis such that the mid-point of the base is at the origin. Find vertices of the triangle.