Hyperbola: Parametric Equation of Hyperbola || Class 11 & IIT JEE

The parametric equation of the hyperbola x²/a² − y²/b² = 1 is x = a sec θ and y = b tan θ, where θ is the parameter. Every value of θ (except θ = π/2 + nπ) gives a valid point on the hyperbola. The identity sec²θ − tan²θ = 1 confirms that these coordinates always satisfy the hyperbola equation

The hyperbola equation involves a subtraction (x²/a² − y²/b² = 1), which matches the identity sec²θ − tan²θ = 1. The ellipse uses addition (x²/a² + y²/b² = 1), matching cos²θ + sin²θ = 1. This structural difference forces the switch from trigonometric to reciprocal trigonometric functions.

For the rectangular hyperbola xy = c², the parametric form is x = ct and y = c/t, where t ≠ 0 is the parameter. This is an algebraic parameter, not an angle. Substituting confirms ct × (c/t) = c². This form is especially useful in JEE Advanced locus and chord-of-contact problems.

At the parametric point (a sec θ, b tan θ) on x²/a² − y²/b² = 1, the tangent equation is (x sec θ)/a − (y tan θ)/b = 1. This is derived by differentiating the hyperbola implicitly and substituting the parametric coordinates. It is one of the most frequently tested results in JEE Main conic section questions.

The standard equation of a hyperbola and its properties are covered in NCERT Class 11 Maths Chapter 11 (Conic Sections). The parametric form itself is introduced as an extension in most coaching syllabi and appears explicitly in JEE Main and JEE Advanced. For full NCERT coverage, refer to the NCERT Solutions for Class 11 Maths.