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

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 _____.

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 points on the hyperbola $x^2-y^2=2$ closest to the point $\left(0, 1\right)$ are

Find the endpoints of the focal chord to the rectangular hyperbola $xy=1$ if the equation of the chord is $y=2\sqrt{2}-x$

Show that a line $y=mx+c$ intersects a rectangular hyperbola $x^2-y^2=a^2$ at two points maximum and the condition for this intersection is $c^2>a^2\left(m^2-1\right)$.
 

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

If the hyperbola $xy={\lambda }^2$ intersects the circle $x^2+y^2=r^2$ in four points $A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)$ and $D(x_4,y_4)$, then value of $\left[\sum _{i=1}^4{x}_i+\prod _{i=1}^4\left(\frac{x_i}{y_i}\right)+\sum _{i=1}^4{y}_i\right]$, where $\left[*\right]$ denotes greatest integer function.

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

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

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?

If $\left(a-3\right)x^2+a{y}^2=9$ represents a rectangular hyperbola, then $a=...$

The value of eccentricity $e$ for a rectangular hyperbola is.....

$PM$ and $PN$ are the perpendiculars from any point on a rectangular hyperbola to its asymptotes. If $Q$ divides $MN$ in the ratio $3:1,$ then the locus of $Q$ is -

If a chord of a rectangular hyperbola, parallel to its conjugate axis subtends angles ${\theta }_1 and {\theta }_2$ at its vertices, then

If the equation $4x^2+k{y}^2=18$ represents a rectangular hyperbola, then $k=$

The equation of a line passing through the centre of a rectangular hyperbola is $x-y-1=0.$ If one of its asymptotes is $3x-4y-6=0,$ the equation of the other asymptote is

Locus of the mid - point of chord of the hyperbola $xy=c^2$ , having slope m, is