If $\theta$ is the acute angle between the curves $x^2+y^2=2020 \sqrt{2}$ and $x^2-y^2=2020$, then $\frac{\sin \theta+\cos \theta}{\tan \theta}$ is equal to

If $(2, a)$ does not lie outside the circles $x^2+y^2=13$ and $x^2+y^2+x-2y=14$,then $a$ lies in

$A$ variable circle passes through the fixed point $A(p, q)$ and touches the $x$-axis. The locus of the other end of the diameter through $A$ is

If the line through the point $P(5,3)$ meets the circle $x^2+y^2-2x-4y+\alpha=0$ at $A(4,2)$ and $B(x_1, y_1)$,then $PA \cdot PB$ is equal to

An equilateral triangle $OAB$ is inscribed in the parabola $y^{2}=4x$ with the vertex $O$ at the origin. Then the minimum distance of the circle having $AB$ as a diameter from the origin is:

Let the line $L: \sqrt{2}x + y = \alpha$ pass through the point of intersection $P$ (in the first quadrant) of the circle $x^2 + y^2 = 3$ and the parabola $x^2 = 2y$. Let the line $L$ touch two circles $C_1$ and $C_2$ of equal radius $2\sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the $y$-axis,then the square of the area of the triangle $PQ_1Q_2$ is equal to: