Basics of Hyperbola

The eccentric angle of the point at which the equation of the tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$ is $3x-y=3\sqrt{3}$ is $30°$

If any point on a hyperbola has the coordinates $(5\sec \varphi ,2\tan \varphi )$. Find the eccentricity of the hyperbola.

If any point on a hyperbola has the coordinates $(3\sec \varphi ,2\tan \varphi )$. Find the eccentricity of the hyperbola.

If any point on a hyperbola has the coordinates $(6\sec \varphi ,4\tan \varphi )$. Find the eccentricity of the hyperbola.

If any point on a hyperbola has the coordinates $(6\sec \varphi ,5\tan \varphi )$. Find the eccentricity of the hyperbola.

Find the equation of the hyperbola whose axes are parallel to the coordinate axes and centre is origin and the length of whose transverse axis is $6$ units and the length of conjugate axis is $4$ units.

The equation $y^2-x^2=81$ represents a rectangular hyperbola.

The equation $y^2-x^2=49$ represents a rectangular hyperbola.

The equation $y^2-x^2=25$ represents a rectangular hyperbola.

The equation $y^2-x^2=9$ represents a rectangular hyperbola.

The equation $y^2-x^2=2$ represents a rectangular hyperbola.

Draw the standard equation of hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$.

Find the equation of the auxiliary circle of the hyperbola $\frac{x^2}{64}-\frac{y^2}{36}=1$.

Find the equation of the auxiliary circle of the hyperbola $4x^2-9y^2=36$.

Find the equation of the hyperbola whose axes are parallel to the coordinate axes and centre is origin and the length of whose transverse axis is $4$ and the length of conjugate axis is $4\sqrt{3}$.

If the eccentricity of a hyperbola is $\frac{5}{4}$, then find the eccentricity of its conjugate.

If the length of latus rectum of a rectangular hyperbola is $18$ unit, find its equation.

For rectangular hyperbola $xy=c^2$, What is the length of transverse axis, length of conjugate axis and length of latus rectum

If the length of latus rectum of a rectangular hyperbola is $6$ unit, find its equation.

Find the length of the latus-rectum of the rectangular hyperbola $x^2-y^2-81=0$.