A straight line through the point $(1, 1)$ meets the $x$-axis at ‘$A$’ and the $y$-axis at ‘$B$’. The locus of the mid-point of $AB$ is
$2xy + x + y = 0$
$x + y - 2xy = 0$
$x + y + 2 = 0$
$x + y - 2 = 0$
Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side a such that $a^{2}+11 a+3\left(m_{2}^{2}+m_{2}^{2}\right)=220$. If one vertex of the square is $(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$, where $\alpha \in\left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos \alpha-\sin \alpha) x +(\sin \alpha+\cos \alpha) y =10$, then $72 \left(\sin ^{4} \alpha+\cos ^{4} \alpha\right)+a^{2}-3 a+13$ is equal to.
Let $m, n$ be real numbers such that $0 \leq m \leq \sqrt{3}$ and $-\sqrt{3} \leq n \leq 0$. The minimum possible area of the region of the plane consisting of points $(x, y)$ satisfying in inequalities $y \geq 0, y-3 \leq m x$, $y -3 \leq nx$, is
The area of the triangle formed by the line $x\sin \alpha + y\cos \alpha = \sin 2\alpha $and the coordinates axes is
Area of the rhombus bounded by the four lines, $ax \pm by \pm c = 0$ is :