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

$A$ triangle $ABC$ lying in the first quadrant has two vertices as $A(1, 2)$ and $B(3, 1)$. If $\angle BAC = 90^{\circ}$ and $\text{ar}(\Delta ABC) = 5\sqrt{5}$ sq. units,then the abscissa of the vertex $C$ is

Given three points $P, Q, R$ with $P(5, 3)$ and $R$ lies on the $x-$ axis. If the 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 the line $3x + 4y - 24 = 0$ intersects the $x-$axis at the point $A$ and the $y-$axis at the point $B$,then the incentre of the triangle $OAB$,where $O$ is the origin,is

If the lines $3x + y - 4 = 0$,$x - \alpha y + 10 = 0$,$\beta x + 2y + 4 = 0$,and $3x + y + k = 0$ represent the sides of a square,then $\alpha \beta (k + 4)^2 = $

The points $(-a,-b), (a, b), (0,0)$ and $(a^{2}, ab)$ where $a \neq 0, b \neq 0$ are always