If $y = m_{1}x + c_{1}$ and $y = m_{2}x + c_{2}$ with $m_{1} \neq m_{2}$ are two common tangents of the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = x$,then the value of $8|m_{1}m_{2}|$ is equal to

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 a tangent is drawn from the focus of the parabola $y^2 = 16x$ to the circle $(x - 6)^2 + y^2 = 2$,find the slope of this tangent.

Let $C_i \equiv x^2 + y^2 = i^2$ for $i = 1, 2, 3$ be three circles. There are $4i$ points on the circumference of each circle $C_i$. If no three of all the points on the three circles are collinear,then the number of triangles that can be formed using these points whose circumcentre does not lie on the origin is:

For the two curves $C_1 : y^2 = 4x$ and $C_2 : x^2 + y^2 - 6x + 1 = 0$,which of the following is true?

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and the circle $x^{2}+y^{2}=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to: