The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt 2 $, is
${x^2} - {y^2} = 16$
${x^2} - {y^2} = 32$
${x^2} - 2{y^2} = 16$
${y^2} - {x^2} = 16$
The equation of the hyperbola whose directrix is $x + 2y = 1$, focus $(2, 1)$ and eccentricity $2$ will be
Let the foci of a hyperbola $\mathrm{H}$ 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 $\mathrm{H}$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $\mathrm{H}$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to :
Locus of foot of normal drawn from any focus to variable tangent of hyperbola $4x^2-9y^2-8x- 18y = 41$ will be
The coordinates of the foci of the rectangular hyperbola $xy = {c^2}$ are
The equation of the tangent to the hyperbola $4{y^2} = {x^2} - 1$ at the point $(1, 0)$ is