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

If the latus rectum of a hyperbola through one focus subtends an angle of $60^{\circ}$ at the other focus,then its eccentricity is

Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$,then $3 \alpha^2+2 \beta^2$ is equal to :

Find the eccentricity of a hyperbola whose latus rectum is $8$ and whose conjugate axis is half the distance between its foci.

Let $P(\frac{\pi}{4}), Q(\frac{5 \pi}{4}), R(\frac{3 \pi}{4}), T(\frac{7 \pi}{4})$ be the points on the hyperbola $x^2-4y^2-4=0$ in the parametric form. Then the area of the quadrilateral $PQRT$ is (in square units) (in $\sqrt{2}$)

Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$,where $l$ and $m$ are co-prime numbers,then $l^2+m^2+n^2$ is equal to . . . . . . .