If the curves $y^2=6x$ and $9x^2+by^2=16$ intersect each other at right angles,then the value of $b$ is

Match the following parametric forms in List-$I$ with their corresponding conic sections in List-$II$:

List-$I$	List-$II$
$(A)$ $\left[\frac{p}{2}\left(t+\frac{1}{t}\right), \frac{q}{2}\left(t-\frac{1}{t}\right)\right]$	$(I)$ parabola
$(B)$ $(p+q \cos \theta, r+q \sin \theta)$	$(II)$ circle
$(C)$ $(p+\lambda^2, q-\lambda)$	$(III)$ ellipse
	$(IV)$ hyperbola

The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ is:

The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\frac{\sqrt{3}}{2}$ and its directrices are $x = \pm \frac{4\sqrt{6}}{3}$. Let $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of $E$, and whose length of latus rectum is equal to the length of minor axis of $E$. Then the distance between the foci of $H$ is :

For some $\theta \in \left(0, \frac{\pi}{2}\right),$ if the eccentricity of the hyperbola $x^{2} - y^{2} \sec^{2} \theta = 10$ is $\sqrt{5}$ times the eccentricity of the ellipse $x^{2} \sec^{2} \theta + y^{2} = 5,$ then the length of the latus rectum of the ellipse is

The circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ intersect at points $A$ and $B$. The equation of the common tangent with a positive slope to the circle and the hyperbola is: