The vertices of the hyperbola $7x^2 - 49y^2 = 343$ are

In a hyperbola,if the length of the transverse axis is twice that of the conjugate axis,then the distance between its directrices is..... units.

For the hyperbola $\frac{x^{2}}{\cos^{2} \alpha} - \frac{y^{2}}{\sin^{2} \alpha} = 1$, which of the following remains fixed when $\alpha$ varies?

Let $PQ$ be a chord of the hyperbola $\frac{x^2}{4} - \frac{y^2}{b^2} = 1$, perpendicular to the $x$-axis such that $OPQ$ is an equilateral triangle, where $O$ is the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$, then the area of the triangle $OPQ$ is:

The locus of a point $P(\alpha, \beta)$ moving under the condition that the line $y = \alpha x + \beta$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is

Let $C$ be the centre of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $P$ be a point on it. If the tangent at $P$ to the hyperbola meets the straight lines $bx-ay=0$ and $bx+ay=0$ respectively in $Q$ and $R$,then $CQ \cdot CR=$