Embibe Experts Solutions for Exercise 11: EXERCISE 13.4

If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is

Chords of an ellipse are drawn through the positive end of the minor axis. Then, their midpoint lies on

If $P\left(\theta \right)$ and $Q\left(\frac{\pi }{2}+\theta \right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then locus of the mid-point of $PQ$ is

In the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1; y=x$ and $3y+2x=0$ are a pair of conjugate diameters. Then the eccentricity of the ellipse is

If the polar with respect to $y^2=4ax$ touches the ellipse $\frac{x^2}{{\mathrm{\alpha }}^2}+\frac{y^2}{{\mathrm{\beta }}^2}=1,$ the locus of its pole is

The equation of the hyperbola whose directrix is $2x+y=1,$ corresponding focus is $(1, 2)$ and eccentricity $\sqrt{3},$ is given by

If the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide, then the value of $b^2$ is

For hyperbola $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\displaystyle {\sin }^2\alpha }}=1$which of the following remains constant with change in '$\alpha$'?