The product of the slopes of the common tangents to the ellipse $\frac{x^2}{32} + \frac{y^2}{8} = 1$ and the parabola $y^2 = 8x$ is:

For $0 < \theta < \pi / 2$,if the eccentricity of the hyperbola $x^2 - y^2 \operatorname{cosec}^2 \theta = 5$ is $\sqrt{7}$ times the eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \theta + y^2 = 5$,then the value of $\theta$ is:

Consider a circle with its centre lying on the focus of the parabola $y^2 = 2px$ such that it touches the directrix of the parabola. Then,a point of intersection of the circle and the parabola is

The tangent at $P$ to a parabola $y^2 = 4ax$ meets the directrix at $U$ and the latus rectum at $V$. Then $\triangle SUV$ (where $S$ is the focus) :

$PQ$ is a normal chord of the parabola $y^2 = 4ax$ at $P$,$A$ being the vertex of the parabola. Through $P$,a line is drawn parallel to $AQ$ meeting the $x$-axis in $R$. Then the length of $AR$ is:

The equation of the line passing through the points of intersection of the parabola $x^2 = 8y$ and the ellipse $\frac{x^2}{3} + y^2 = 1$ is