$2x - 3y + 1 = 0$ and $4x - 5y - 1 = 0$ are the equations of two diameters of the circle $S \equiv x^2 + y^2 + 2gx + 2fy - 11 = 0$. $Q$ and $R$ are the points of contact of the tangents drawn from the point $P(-2, -2)$ to this circle. If $C$ is the centre of the circle $S = 0$,then the area (in square units) of the quadrilateral $PQCR$ is

If $(\alpha, \beta)$ is a point on the circle whose centre is on the $x$-axis and which touches the line $x + y = 0$ at $(2, -2)$,then the greatest value of $\alpha$ is

$A$ line drawn through the point $P(4, 7)$ cuts the circle $x^2 + y^2 = 9$ at the points $A$ and $B$. Then $PA \cdot PB$ is equal to

Let circle $C$ be the image of $x^2+y^2-2x+4y-4=0$ in the line $2x-3y+5=0$. Let $A$ be the point on $C$ such that $OA$ is parallel to the $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$,with $\beta < 4$,lies on $C$ such that the length of the arc $AB$ is $(1/6)^{\text{th}}$ of the perimeter of $C$,then $\beta - \sqrt{3}\alpha$ is equal to

$A$ circle touches the parabola $y^2=4x$ at $(1,2)$ and also touches its directrix. The $y$-coordinate of the point of contact of the circle and the directrix is

If $\theta$ is the acute angle of intersection at a real point of intersection of the circle $x^2 + y^2 = 5$ and the parabola $y^2 = 4x$,then $\tan \theta$ is equal to