Hyperbola

An ellipse passes through the foci of the hyperbola, $9x^2-4y^2=36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\dfrac12$, then which of the following points does not lie on the ellipse ?

Let $P(3\sec\theta,2\tan\theta)$ and $Q(3\sec\phi,2\tan\phi)$ where $\theta+\phi=\dfrac\pi2$, be two distinct points on the hyperbola $\dfrac{x^2}9-\dfrac{y^2}4=1$. Then the ordinate of the point of intersection of the normals at P and Q is :

The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola $\dfrac{x^2}4-\dfrac{y^2}5=1$, meets $x$ - axis and $y$ - axis at A and B respectively. Then $(OA)^2-(OB)^2$, where O is the origin, equals :

Tangents are drawn to the hyperbola $\dfrac{x^2}9-\dfrac{y^2}4=1$, parallel to the straight line $2x-y=1$. The points of contact of the tangents on the hyperbola are

Let the eccentricity of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4y^2=4$. If the hyperbola passes through a focus of the ellipse, then

Let $P(6,3)$ be a point on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. If the normal at the point $P$ intersects the $x$-axis at $(9,0)$, then the eccentricity of the hyperbola is

Equation of the circle with AB as its diameter is

The line $2x+y=1$ is tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is

Match the conics in Column I with the statements/expressions in Column II.

An ellipse intersects the hyperbola $2x^2-2y^2=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

Consider a branch of the hyperbola
$$x^2-2y^2-2\sqrt2x-4\sqrt2y-6=0$$
with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $ABC$ is

A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2+4y^2=12$. Then its equation is

Match the statements in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.