IIT JEE Formulas for Rectangular Hyperbolas
Shifting of origin of the rectangular hyperbola is a very important concept for students, suppose we take a rectangular hyperbola and the coordinate of any point is A(x, y) and its origin is rotated anticlockwise by π/4 then the in new coordinate system the point A is transformed to B(X, Y) where,
- X = x.cosα – y.sinα = x.cos(π/4) + y.sin(π/4) = (x – y)/√(2)…(i)
- Y = x.sinα + y.cosα = x.sin(π/4) + y.cos(π/4) = (x + y)/√(2)…(ii)
Now equation of the rectangular hyperbola is,
X2 – Y2 = a2
⇒ {(x – y)/√(2)}2 – {(x + y)/√(2)}2 = a2
⇒ (x2 + y2 – 2xy)/2 – (x2 + y2 + 2xy)/2 = a2
⇒ -4xy/2 = a2
⇒ xy = a2/-2
Let, c = -1/2a2 then equation becomes
xy = constant
Now the various formulas for the rectangular hyperbola xy = c2 with parameter ‘t’ and any point (ct, c/t) are,
Foci |
(±√(2)c, ±√(2)c) |
Eccentricity |
√(2)c |
Transverse Axis |
2√(2)c |
Directrices |
x + y = ±√(2)c |
Asymptotes |
|
People Also Read:
Rectangular Hyperbola
Rectangular Hyperbola is a hyperbola in which the transverse and conjugate axes are equal. i.e. in the case of rectangular hyperbola a = b = 1. The asymptote of the rectangular hyperbola is y = ±x. Also, the asymptotes of a rectangular hyperbola are perpendicular.
In this article, we will explore the rectangular hyperbola in depth along with its standard equation, eccentricity, asymptotes, and parametric equation.
Table of Content
- What is a Rectangular Hyperbola?
- Rectangular Hyperbola Shape
- Rectangular Hyperbola Equation
- Parametric Equation of Rectangular Hyperbola
- Rectangular Hyperbola Graph
- Rectangular Hyperbola Formulas
- Properties of Rectangular Hyperbola
- IIT JEE Formulas for Rectangular Hyperbolas
- Examples on Rectangular Hyperbola
- Practice Questions on Rectangular Hyperbola