The locus of the point of instruction of the lines $\sqrt 3 x – y – 4 \sqrt 3 t = 0$  $\&$  $\sqrt 3tx + ty – 4\sqrt 3 = 0$  (where $ t$  is a parameter) is a hyperbola whose eccentricity is

Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P (4,1)$ to the hyperbola $H: \frac{y^2}{25}-\frac{x^2}{16}=1$. If $Q$ is the point from which the tangents drawn to $H$ have slopes $\left| m _1\right|$ and $\left| m _2\right|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x$ axis, then $\frac{(P Q)^2}{\alpha \beta}$ is equal to $…………$.

Tangents are drawn to the hyperbola $4{x^2} – {y^2} = 36$ at the points $P$ and $Q.$ If these tangents intersect at the point $T(0,3)$ then the area (in sq. units) of $\Delta PTQ$ is :

Let the equation of two diameters of a circle $x ^{2}+ y ^{2}$ $-2 x +2 fy +1=0$ be $2 px – y =1$ and $2 x + py =4 p$. Then the slope $m \in(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to $……$

If the eccentricity of a hyperbola $\frac{{{x^2}}}{9} – \frac{{{y^2}}}{{{b^2}}} = 1,$ which passes through $(K, 2),$ is $\frac{{\sqrt {13} }}{3},$ then the value of $K^2$ is