Rectangular Hyperbola

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

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 :

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

The length of the latus rectum of the rectangular hyperbola $xy=32$ is

The equation $y^2-x^2=64$ represents a rectangular hyperbola.

The equation $y^2-x^2=16$ represents a rectangular hyperbola.

The equation $y^2-x^2=4$ represents a rectangular hyperbola.

The equation $y^2-x^2=1$ represents a rectangular hyperbola.

The equation $y^2-x^2+2x-1=0$ represents a rectangular hyperbola.

 What type of conic is represented by the equation $x^2-y^2=9?$  What is its eccentricity?

Show that the eccentricity of any rectangular hyperbola is $\sqrt{2}$.

Eccentricity of the conic represented by curve $x^2-y^2=4x-2y$ is

If the equation $4x^2+k{y}^2=18$ represents a rectangular hyperbola, find the value of it.

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

If $P,Q,R,S$ are the $4$ points of intersection of the circle $x^2+y^2-6x-4y-3=0$ with the curve $y=\frac{2x+10}{x-3}$ with center as $C$, then the value of ${\left(\mathrm{\text{CP}}\right)}^2+{\left(\mathrm{\text{CQ}}\right)}^2{+\left(\mathrm{\text{CR}}\right)}^2+{\left(\mathrm{\text{CS}}\right)}^2$ is $k$, then $\frac{k}{8}$=

In the figure, shaded rectangles have area $A \& B$, then  $\frac{B}{A}=$

From the following option which is true if a circle cuts rectangular hyperbola $xy=1$ in the points $\left(x_r, y_r\right), r=\mathrm{1,} 2, 3, 4.$ 

Consider a point $(a, \alpha )$ lying on a rectangular hyperbola $xy=c^2$ such that $c$is purely imaginary. If $a$ is a positive number then $\alpha$ is ____

The curve $\mathrm{xy}=\mathrm{c} , \left(\mathrm{c} >0\right)$, and the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=1$ touch at two points. Then the distance bewteen the points of contacts is.

If the rectangular hyperbola is $x^2-y^2=64$ . Then, which of the following is not correct?