If the center,vertex,and focus of a hyperbola are $(0, 0)$,$(4, 0)$,and $(6, 0)$ respectively,then the equation of the hyperbola is:

$P(\theta)$ 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 $\theta =$

The product of the lengths of the perpendiculars from any point on the hyperbola $x^2-y^2=16$ to its asymptotes is

$A$ square $ABCD$ has all its vertices on the curve $x^{2}y^{2}=1$. The midpoints of its sides also lie on the same curve. Then,the square of the area of $ABCD$ is

If $P(\frac{\pi}{4})$ and $Q(\frac{3 \pi}{4})$ are two points on the hyperbola $4 x^2-y^2-8 x-2 y-13=0$ in parametric form,then the distance between $P$ and $Q$ 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?