If $ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ represents a joint equation of the directrices of the hyperbola $16x^{2}-9y^{2}=144$,then $g+f-c=$

Let $\lambda x - 2y = \mu$ be a tangent to the hyperbola $a^{2}x^{2} - y^{2} = b^{2}$. Then $\left(\frac{\lambda}{a}\right)^{2} - \left(\frac{\mu}{b}\right)^{2}$ is equal to

$A$ double ordinate $PQ$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is such that $\Delta OPQ$ is equilateral, where $O$ is the centre of the hyperbola. Then the eccentricity $e$ satisfies the relation:

Let $e$ be the eccentricity of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$. If $\frac{1}{e}$ is the eccentricity of a hyperbola,then the eccentricity of its conjugate hyperbola is

The locus of the midpoints of the chord of the circle $x^{2}+y^{2}=25$ which is tangent to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ is

The product of the lengths of the perpendiculars from any point on the hyperbola $x^2 - y^2 = 8$ to its asymptotes is