Rectangular Hyperbola Having Coordinate Axes as Asymptotes

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

Prove that the diameter of one hyperbola is conjugate to its reflection in the asymptote, which is a diameter of the other hyperbola.

Find the conjugate diameters of a rectangular hyperbola $x^2-y^2=a^2$

Find the diameter of the rectangular hyperbola $x^2-y^2=a^2$ passes through its centre.

Find the point of intersection of the normals to a rectangular hyperbola $xy=c^2$ in parametric form.

Find the equation of the normals from a point $P\left(0, 1\right)$ to the rectangular hyperbola $xy=4$ in the separate form.

Find the intersection points of a line $2y=x$ and the rectangular hyperbola $x^2-y^2=1$.

Find the condition of tangency and point of tangency, if line $y=mx+c$ touches the rectangular hyperbola $x^2-y^2=a^2$.

The equation of the director circle to the rectangular hyperbola $x^2-y^2=a^2$ is

Find the combined equation of the pair of tangents drawn from a point $P\left(1, 0\right)$ to the Rectangular Hyperbola $xy=4$ and write the separate equation of lines represented by the combined equation.

Find the combined equation of the pair of tangents drawn from a point $P\left(x_1, y_1\right)$ to the Rectangular Hyperbola $xy=c^2$.

 If the tangents drawn from a point  $P\left(1, 2\right)$ to the rectangular hyperbola $x^2-y^2=1$ touch the rectangular hyperbola at $Q \& R$, then find the equation of the chord of contact $QR$.

 If the tangents drawn from a point  $P\left(1, 2\right)$ to the rectangular hyperbola $x^2-y^2=1$ touch the rectangular hyperbola at $Q \& R$, then find the length of the chord of contact $QR$.

Prove that the perpendicular focal chords of a rectangular hyperbola are equal.

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 $c^2>a^2\left(m^2-1\right)$.
 

The number of normal(s) of a rectangular hyperbola which can touch its conjugate is equal to

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.

The length of the latusrectum of the conic $y=\frac{4}{x}$ is

Circles are drawn on the chords of the rectangular hyperbola $xy=4$ parallel to the line $y=x$ as diameters. All such circles pass through two fixed points whose coordinates are

The normal to the rectangular hyperbola $xy=c^2$ at the point ‘t’ meets the curve again at a point ‘t’, such that