The graph of the conic $ x^2 – (y – 1)^2 = 1$  has one tangent line with positive slope that passes through the origin. the point of tangency being $(a, b). $ Then  Length of the latus rectum of the conic is

Tangents are drawn from any point on hyperbola $4x^2 -9y^2 = 36$ to the circle $x^2 + y^2 = 9$ . If locus of midpoint of chord of contact is $\left( {\frac{{{x^2}}}{9} – \frac{{{y^2}}}{4}} \right) = \lambda {\left( {\frac{{{x^2} + {y^2}}}{9}} \right)^2}$ , then $\lambda $ is 

For the Hyperbola ${x^2}{\sec ^2}\theta – {y^2}cose{c^2}\theta = 1$ which of the following remains constant when $\theta $ varies $= ?$

A tangent to the hyperbola $\frac{{{x^2}}}{4} – \frac{{{y^2}}}{2} = 1$ meets $x-$ axis at $P$ and $y-$ axis at $Q$. Lines $PR$ and $QR$ are drawn such that $OPRQ$ is a rectangle (where $O$ is the origin). Then $R$ lies on

If $2 x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$, then which of the following $CANNOT$ be sides of a right angled triangle?

$[A]$ $2 a, 4,1$   $[B]$ $2 a, 8,1$   $[C]$ $a, 4,1$    $[D]$ $a, 4,2$