The differential equation $\frac{{dx}}{{dy}}= \frac{{3y}}{{2x}}$ represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity :

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

The graph of the conic $ x^2 - (y - 1)^2 = 1$  has one tangent line with positive slope that passes through the origin. the point of tangency being $(a, b). $ Then  Length of the latus rectum of the conic is

On a rectangular hy perbola $x^2-y^2= a ^2, a >0$, three points $A, B, C$ are taken as follows: $A=(-a, 0) ; B$ and $C$ are placed symmetrically with respect to the $X$-axis on the branch of the hyperbola not containing $A$. Suppose that the $\triangle A B C$ is equilateral. If the side length of the $\triangle A B C$ is $k a$, then $k$ lies in the interval

Consider a hyperbola $\mathrm{H}$ having centre at the origin and foci and the $\mathrm{x}$-axis. Let $\mathrm{C}_1$ be the circle touching the hyperbola $\mathrm{H}$ and having the centre at the origin. Let $\mathrm{C}_2$ be the circle touching the hyperbola $\mathrm{H}$ at its vertex and having the centre at one of its foci. If areas (in sq. units) of $\mathrm{C}_1$ and $\mathrm{C}_2$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of $\mathrm{H}$ is

The eccentricity of the conjugate hyperbola of the hyperbola ${x^2} - 3{y^2} = 1$, is