The product of the lengths of the perpendiculars drawn from the foci to any tangent to the hyperbola $x^{2} - \frac{y^{2}}{4} = 1$ is:

The sound of a cannon firing is heard one second later at position $B$ than at position $A$. If the speed of sound is uniform,then

If the angle between the asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan^{-1}\left(\frac{b}{a}\right) = 2 \tan^{-1}\left(\frac{2}{3}\right)$ and $a^2-b^2=45$,then $ab=$

If $ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ represents a joint equation of the directrices of the hyperbola $16x^{2}-9y^{2}=144$,then $g+f-c=$

What kind of hyperbola does the equation $9x^2 - 16y^2 - 18x + 32y - 151 = 0$ represent?

If the eccentricity of the hyperbola $x^2 - y^2 \sec^2 \alpha = 5$ is $\sqrt{3}$ times the eccentricity of the ellipse $x^2 \sec^2 \alpha + y^2 = 25$,then a value of $\alpha$ is: