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 line $ax$ + $by$ = $1$ is normal to the hyperbola $\frac{{{x^2}}}{{{p^2}}} - \frac{{{y^2}}}{{{q^2}}} = 1$ then $\frac{{{p^2}}}{{{a^2}}} - \frac{{{q^2}}}{{{b^2}}} = 1$ is equal to (where $a$,$b$,$p$, $q \in {R^ + })$-

The foci of a hyperbola are $( \pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2 x+3 y=6$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$ - and $y$-axes are $a$ and $b$ respectively, then $|6 a|+|5 b|$ is equal to $..........$.

The length of transverse axis of the parabola $3{x^2} - 4{y^2} = 32$ is

The locus of the centroid of the triangle formed by any point $\mathrm{P}$ on the hyperbola $16 \mathrm{x}^{2}-9 \mathrm{y}^{2}+$ $32 x+36 y-164=0$, and its foci is:

If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercept  $a_1, a_2$ on $x-$ axis and $b_1, b_2$ on $y-$ axis, then $(a_1a_2 + b_1b_2)$ is