The length of the tangent drawn from the mid-point of the line joining the origin and the point $(4, -4)$ to the circle $2x^2 + 2y^2 - y = 0$ is

If the tangent at the point $P$ on the circle $x^2+y^2+6x+6y=2$ meets the straight line $5x-2y+6=0$ at a point $Q$ on the $Y$-axis,then the length of $PQ$ is

Two tangents to the circle $x^{2}+y^{2}=4$ at the points $A$ and $B$ meet at $M(-4,0)$. The area of the quadrilateral $MAOB$, where $O$ is the origin, is

Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $(\alpha, \frac{1}{2})$,then a value of $\alpha$ is

Let a circle of radius $4$ and the ellipse $15x^2 + 19y^2 = 285$ be concentric. What is the angle that the common tangents make with the minor axis of the ellipse?

The equation of the common tangent touching the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4x$ above the $x$-axis is