Here, you will learn what is rectangular hyperbola and parametric equation of rectangular hyperbola with example.
Letโs begin โ
Rectangular Hyperbola
The particular kind of the hyperbola in which the length of itโs transverse axis and conjugate axis are equal is called rectangular hyperbola. The eccentricity of the rectangular hyperbola is \(\sqrt{2}\) and the length of itโs latus rectum is equal to itโs 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 โ {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 โ \(c\over t\) = \(t^2\)(x โ 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!
