If a circle intersects the hyperbola $xy = 1$ at four points $(x_r, y_r)$ for $r = 1, 2, 3, 4$,then:

Let the origin be the centre,$(\pm 3, 0)$ be the foci,and $\frac{3}{2}$ be the eccentricity of a hyperbola. Then the line $2x - y - 1 = 0$

Let $x^2+y^2=16$ be the equation of the auxiliary circle of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and let $(4 \sqrt{2}, 3)$ be a point on the hyperbola. Then,the eccentricity of the hyperbola is

If $\theta$ is the acute angle between the tangents drawn from the point $(2,3)$ to the hyperbola $5x^2-6y^2-30=0$,then $\tan \theta=$

Let $A(-1, 0)$ and $B(2, 0)$ be two points. $A$ point $M$ moves in the plane in such a way that $\angle MBA = 2 \angle MAB$. Then, the point $M$ moves along

Let $P(3 \sec \theta, 2 \tan \theta)$ and $Q(3 \sec \phi, 2 \tan \phi)$ be two points on the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ such that $\theta + \phi = \frac{\pi}{2}$ where $0 < \theta, \phi < \frac{\pi}{2}$. Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is: