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 $......$

Eccentricity of conjugate hyperbola of $16x^2 - 9y^2 - 32x - 36y - 164 = 0$ will be-

The locus of a point $P (h, k)$ such that the line $y = hx + k$ is tangent to $4x^2 - 3y^2 = 1$ , is a/an

The length of the latus rectum of the hyperbola $25x^2 -16y^2 = 400$ is -

Consider a branch of the hyperbola $x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $\mathrm{C}$ is the focus of the hyperbola nearest to the point $\mathrm{A}$, then the area of the triangle $\mathrm{ABC}$ is

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