Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 3 \sqrt{5}, 0)$,the latus rectum is of length $8$.

If the eccentricity of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,passing through $(6, 4\sqrt{3})$,satisfies $15(e^2 + 1) = 34e$,then the length of the latus rectum of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{2(a^2 + 1)} = 1$ is:

The line $21x + 5y = k$ touches the hyperbola $7x^2 - 5y^2 = 232$. Then $k =$

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 $y_1 > 0$

Let $x^2+y^2=16$ be the equation of the auxiliary circle of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and let $(4 \sqrt{2}, 3)$ be a point on the hyperbola. Then,the eccentricity of the hyperbola is

Find the coordinates of the foci and the vertices,the eccentricity,and the length of the latus rectum of the hyperbola: $y^{2}-16x^{2}=16$.