$A$ rhombus is inscribed in the region common to the two circles $x^2 + y^2 - 4x - 12 = 0$ and $x^2 + y^2 + 4x - 12 = 0$,with two of its vertices on the line joining the centres of the circles. The area of the rhombus is:

Consider a pair of circles $(|x| - 1)^2 + y^2 = 1$. Ram is moving along the circle centered at $(1, 0)$ in the clockwise direction at a rate of $2 \ m/s$,and Shyam is moving along the circle centered at $(-1, 0)$ in the anticlockwise direction at a rate of $1 \ m/s$. If Ram and Shyam start their journey from the origin $(0, 0)$,then the rate of change of the distance between Ram and Shyam at the instant when Ram crosses the $x$-axis for the first time is:

If $x+y-1=0$ and $2x-y+1=0$ are conjugate lines with respect to a circle $x^2+y^2-4x+2fy-1=0$,then $f=$

$A$ circle $S \equiv x^2+y^2+2gx+2fy+6=0$ cuts another circle $x^2+y^2-6x-6y-6=0$ orthogonally. If the angle between the circles $S=0$ and $x^2+y^2+6x+6y+2=0$ is $60^{\circ}$,then the radius of the circle $S=0$ is

Let $P(3 \cos \alpha, 2 \sin \alpha)$, $\alpha \neq 0$, be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$, $Q$ be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$, and $R$ be a point on the line $x + y = 5$ such that the centroid of the triangle $PQR$ is $(2 + \cos \alpha, 3 + \frac{2}{3} \sin \alpha)$. Then the sum of the ordinates of all possible points $R$ is:

If the two circles,$x^2 + y^2 + 2g_1x + 2f_1y = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y = 0$ touch each other,then: