The locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^2$  on a variable tangent is :

The values of parameter $'a'$ such that the line $\left( {{{\log }_2}\left( {1 + 5a – {a^2}} \right)} \right)x – 5y – \left( {{a^2} – 5} \right) = 0$ is a normal to the curve $xy = 1$ , may lie in the interval

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to

If $(0,\; \pm 4)$ and $(0,\; \pm 2)$ be the foci and vertices of a hyperbola, then its equation is

The chord $ PQ $ of the rectangular hyperbola $xy = a^2$  meets the axis of $x$ at $A ; C $ is the mid point of $  PQ\ \& 'O' $ is the origin. Then the $ \Delta ACO$  is :