If $\frac{x}{a} + \frac{y}{b} = 1$ is a tangent to the curve $x = Kt, y = \frac{K}{t}, K > 0$,then

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 $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ be two points such that $\theta+\phi=\frac{\pi}{2}$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$,then $k=$

The equation $\frac{1}{r} = \frac{1}{8} + \frac{3}{8} \cos \theta$ represents:

If $P$ is a point on the hyperbola $16x^2 - 9y^2 = 144$ whose foci are $S_1$ and $S_2$,then $|PS_1 - PS_2| = $

The equation of the line passing through the points $\left(ct_1, \frac{c}{t_1}\right)$ and $\left(ct_2, \frac{c}{t_2}\right)$ is