The $X$ and $Y$ intercepts of the tangent to the hyperbola $\frac{x^2}{20}-\frac{y^2}{5}=1$ which is perpendicular to the line $4x+3y=7$,are respectively

The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt{2}$ is:

If the tangent at the point $(2 \sec \phi, 3 \tan \phi)$ of the hyperbola $\frac{x^2}{4} - \frac{y^2}{9} = 1$ is parallel to $3x - y + 4 = 0$,then the value of $\phi$ is ............ $^o$.

Consider the hyperbola $H : x^2-y^2=1$ and a circle $S$ with center $N(x_2, 0)$. Suppose that $H$ and $S$ touch each other at a point $P(x_1, y_1)$ with $x_1 > 1$ and $y_1 > 0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If $(l, m)$ is the centroid of the triangle $\triangle PMN$,then the correct expression$(s)$ is(are):
$(A) \frac{dl}{dx_1} = 1 - \frac{1}{3x_1^2}$ for $x_1 > 1$
$(B) \frac{dm}{dx_1} = \frac{x_1}{3\sqrt{x_1^2-1}}$ for $x_1 > 1$
$(C) \frac{dl}{dx_1} = 1 + \frac{1}{3x_1^2}$ for $x_1 > 1$
$(D) \frac{dm}{dy_1} = \frac{1}{3}$ for $y_1 > 0$

If $A$ and $B$ are two points in a plane,such that $|PA - PB| = \text{constant}$,then the locus of $P$ is

The foci of the hyperbola $2x^2 - 3y^2 = 5$ are