The locus of the point of intersection of the lines $bxt - ayt = ab$ and $bx + ay = abt$ is

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$. Its equation is

The locus of the point of intersection of tangents at the ends of a normal chord of the hyperbola $x^2 - y^2 = a^2$ is

The locus of a point of intersection of two lines $x \sqrt{3}-y=k \sqrt{3}$ and $\sqrt{3} k x+k y=\sqrt{3}$,where $k \in R$,describes:

What is the area of the triangle formed by the lines $x - y = 0$,$x + y = 0$,and any tangent to the hyperbola $x^{2} - y^{2} = a^{2}$?

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 $y_1 > 0$