If $5x + 9 = 0$ is the directrix of the hyperbola $16x^2 - 9y^2 = 144,$ then its corresponding focus is

Let the tangent drawn to the parabola $y^2 = 24x$ at the point $(\alpha, \beta)$ be perpendicular to the line $2x + 2y = 5$. Then the normal to the hyperbola $\frac{x^2}{\alpha^2} - \frac{y^2}{\beta^2} = 1$ at the point $(\alpha + 4, \beta + 4)$ does $NOT$ pass through which of the following points?

$P(\theta)$ is a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{9} = 1$,$S$ is its focus lying on the positive $X$-axis and $Q = (0, 1)$. If $S Q = \sqrt{26}$ and $S P = 6$,then $\theta =$

If the coordinates of a point are given by the equations $x = b \sec \phi$ and $y = a \tan \phi$,then its locus 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

$A$ hyperbola passes through a focus of the ellipse $\frac{x^2}{169}+\frac{y^2}{25}=1$. Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse. The product of their eccentricities is $1$. Then,the equation of the hyperbola is