The circumcenter of the triangle formed by the lines $xy + 2x + 2y + 4 = 0$ and $x + y + 2 = 0$ is

$L \equiv 7x - y + 8 = 0$ is one of the diagonals of a square for which $(-4, 5)$ and $(3, 4)$ are two vertices. Find the coordinates of the two vertices lying on the diagonal $L = 0$.

Suppose $ABCD$ $(AB \parallel CD)$ is a trapezium such that the diagonals $AC$ and $BD$ bisect the angles $\angle DAB$ and $\angle CBA$,respectively. Then

Let $B$ and $C$ be two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on the line $y-2x=2$ such that $\triangle ABC$ is an equilateral triangle. Then,the area of the $\triangle ABC$ is

If $A(6, 3)$,$B(-3, 5)$,$C(4, -2)$,and $D(x, 3x)$ are four points. If the ratio of the area of $\Delta DBC$ and $\Delta ABC$ is $1 : 2$,then the value of $x$ is:

The incentre of the triangle formed by the lines $x+y=1$,$x=1$,and $y=1$ is