Here, you will learn what is rectangular hyperbola and parametric equation of rectangular hyperbola with example.

Let’s begin –

## Rectangular Hyperbola

The particular kind of the hyperbola in which the length of it’s transverse axis and conjugate axis are equal is called rectangular hyperbola. The **eccentricity of the rectangular hyperbola** is \(\sqrt{2}\) and the length of it’s latus rectum is equal to it’s transverse axis or conjugate axis.

The equation of rectangular hyperbola is xy = \(c^2\).

## Parametric Equation of Rectangular Hyperbola

The Rectangular hyperbola referred to its asymptotes as axis of coordinates.

The parametric equation of the rectangular hyperbola xy = \(c^2\) with parametric representation is x = ct, y = c/t, t \(\in\) R – {0}.

## Basic Definitions of Rectangular Hyperbola :

For the hyperbola, xy = \(c^2\)

(i) Vertices : (c, c) & (-c, -c).

(ii) Foci : (\(\sqrt{2c}, \sqrt{2c}\)) & (\(-\sqrt{2c}, -\sqrt{2c}\))

(iii) Directrices : x + y = \(\pm \sqrt{2c}\)

(iv) Latus rectum : l = \(2\sqrt{2c}\) = T.A = C.A

**Note :**

(a) The equation of chord joining the points \(P(t_1)\) & \(Q(t_2)\) is x + \(t_1t_2\)y = c(\(t_1+t_2\)) with slope,

m = \(-1\over t_1t_2\).

(b) The equation of the tangent to rectangular hyperbola in point form at P(\(x_1, y_1\)) is \(x\over x_1\) + \(y\over y_1\) = 2 & in parametric form at P(t) is \(x\over t\) + ty = 2c.

(c) Equation of normal in parametric form is y – \(c\over t\) = \(t^2\)(x – ct).

(d) The equation of chord with a given middle point as (h, k) is kx + hy = 2hk.

Example : Find the parametric representation and equation of tangent at the point (1, 2) to the rectangular hyperbola xy = 2.

Solution : Since, the equation of rectangular hyperbola is xy = 2.

Comparing with the equation of rectangular hyperbola xy = \(c^2\).

we get c = \(\sqrt{2}\)

Equation in parametric representation x = ct and y = c/t.

\(\implies\) x = \(\sqrt{2}\)t and y = \(\sqrt{2}\)/t

Equation of the tangent to rectangular hyperbola in point form at P(\(x_1, y_1\)) is \(x\over x_1\) + \(y\over y_1\) = 2

\(\implies\) \(x\over 1\) + \(y\over 2\) = 2

Hope you learnt what is the equation of rectangular hyperbola, learn more concepts of hyperbola and practice more questions to get ahead in the competition. Good luck!