Hyperbola

Let $H_n:\frac{x^2}{1+n}-\frac{y^2}{3+n}=1, n\in \mathrm{\mathbb{N}}$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H_k$ is a rational number. If $l$ is the length of the latus rectum of $H_k$, then $21l$ is equal to

Let $\mathrm{R}$ be a rectangle given by the lines $x=0, x=2$, $y=0$ and $y=5$. Let $\mathrm{A}\left(\alpha ,0\right)$ and $\mathrm{B}\left(0,\beta \right), \alpha \in \left[0,2\right]$ and $\beta \in \left[0,5\right]$, be such that the line segment $AB$ divides the area of the rectangle $R$ in the ratio $4:1$. Then, the mid-point of $AB$ lies on a

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is _____.

If the line $2x+y+k=0$ is tangent to the hyperbola $\frac{x^2}{6}-\frac{y^2}{8}=1$ then find the value of $k$.

A rectangle is drawn by lines $x=0, x=2, y=0 \& y=5$. Point $A$ and $B$ lie on coordinate axes. If the line $AB$ divides the area of rectangle in $4:1$, then the locus of the midpoint of $AB$ is :

Find the angle between the asymptotes of hyperbola $\frac{x^2}{4}-\frac{y^2}{16}=1$

Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbola $y^2-16x^2=16$.

If the portion of the asymptote between the centre and the tangent at the vertex of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ in the third quadrant is cut by the line $y+\lambda (x+a)=0, \lambda$ being parameter, then

Find the equation of hyperbola whose

$c\equiv \left(0,0\right); v\equiv \left(\pm 4,0\right); f\equiv \left(\pm 6,0\right)$

Let $e_1$ and $e_2$ are eccentricity of two orthogonally hyperbolas, then find the relation between $e_1$ and $e_2.$

tangents are drawn from $(0,\frac{-9}{4})$ to $x^2-y^2=9$,find point of contact?

For the hyperbola $\frac{x^2}{{\cos }^2\alpha }-\frac{{\displaystyle y^2}}{{\displaystyle {\sin }^2\alpha }}=1,e=\frac{2}{\sqrt{3}}$ and $\frac{3\mathrm{\pi }}{2}<\alpha <2\mathrm{\pi }$, $\alpha$ equals 

If $P$ and $Q$ lie on $x^2-y^2=9$ such that area of isosceles triangle $PQR$ where $PR=QR$ is $10$ sq. units and $R\equiv \left(0,-6\right)$ . Find coordinate of $P$.

If $\frac{x^2}{k-\frac{5}{2}}+\frac{y^2}{\frac{7}{3}-k}=1$ ($k$ is a real number) represents a hyperbola, then the set of all values of $k$ is

Let $a>0, b>0$. Let e and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Let $e^{\text{'}}$ and $l$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If ${\mathrm{e}}^2=\frac{11}{14}l$ and ${\left(e^{\text{'}}\right)}^2=\frac{11}{8}{l}^{\text{'}}$, then the value of $77a+44b$ is equal to :

Find the foci of hyperbola $9{\mathrm{x}}^2-16{\mathrm{y}}^2=144$.

If $P(x_1,y_1)$ is a point on the hyperbola $x^2-y^2=a^2$, then $SP·S^{\text{'}}P=$

Find the equation of the hyperbola where foci are $\left(0, \pm 12\right)$ and the length of the latus rectum is $36$.

Find the equation of the hyperbola with foci $\left(0, \pm 3\right)$ and vertices $\left(0, \pm \frac{\sqrt{11}}{2}\right)$.

Find the coordinates of the foci and the vertices, the eccentricity, the length of the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=1$.