Mastering Parametric Equations of a Hyperbola for Competitive Exams

Welcome to this module on Parametric Equations of a Hyperbola, a crucial topic for JEE Mathematics. Understanding parametric forms allows us to represent points on a hyperbola using a single variable, simplifying many calculations and problem-solving approaches. This section will guide you through the fundamental concepts and their application.

Understanding the Standard Hyperbola

Before diving into parametric forms, let's recall the standard equation of a hyperbola centered at the origin. For a hyperbola with its transverse axis along the x-axis, the equation is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ . If the transverse axis is along the y-axis, the equation is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ . Here, 'a' is the semi-transverse axis and 'b' is the semi-conjugate axis.

What is the standard equation of a hyperbola with its transverse axis along the x-axis?

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Deriving Parametric Equations

Consider the standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ . We can express any point $(x, y)$ on this hyperbola in terms of a parameter, typically denoted by $\theta$ or $t$ . A common parametric representation is derived by setting $x = a \sec \theta$ and $y = b \tan \theta$ . Let's verify this:

Substituting these into the standard equation: $\frac{(a \sec \theta)^2}{a^2} - \frac{(b \tan \theta)^2}{b^2} = \frac{a^2 \sec^2 \theta}{a^2} - \frac{b^2 \tan^2 \theta}{b^2} = \sec^2 \theta - \tan^2 \theta = 1$ . This confirms that the parametric equations $x = a \sec \theta$ and $y = b \tan \theta$ represent points on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ .

Parametric equations simplify hyperbola analysis.

Parametric equations for a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are $x = a \sec \theta$ and $y = b \tan \theta$ . This form is useful for solving problems involving tangents, normals, and locus.

The parameter $\theta$ can range from $0$ to $2\pi$ (excluding $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ where $\sec \theta$ is undefined). For the hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ , the parametric equations are $x = b \tan \theta$ and $y = a \sec \theta$ . Another common parametric form for $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $x = a \cosh t$ and $y = b \sinh t$ , where $t$ is any real number. This form is particularly useful when dealing with the real branch of the hyperbola.

Visualizing the parametric representation of a hyperbola helps understand how the parameter $\theta$ traces the curve. For $x = a \sec \theta$ and $y = b \tan \theta$ , as $\theta$ varies, the point $(x, y)$ moves along the hyperbola. The parameter $\theta$ is related to the angle formed by the line connecting the origin to a point on the hyperbola and the conjugate axis. Specifically, if we consider the auxiliary circle and the tangent at a point, $\theta$ can be interpreted geometrically.

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Key Properties and Applications

Parametric equations are invaluable for finding the equation of a tangent or normal to a hyperbola at a given point. For instance, the tangent at $(a \sec \theta, b \tan \theta)$ to $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is given by $\frac{x}{a} \sec \theta - \frac{y}{b} \tan \theta = 1$ . This simplifies complex geometric problems into algebraic manipulations of the parameter.

Remember that the parameter $\theta$ in $x = a \sec \theta, y = b \tan \theta$ is not the same as the angle in polar coordinates.

Example Problem

Find the equation of the tangent to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point where $\theta = \frac{\pi}{3}$ . Here, $a^2 = 16 \implies a = 4$ and $b^2 = 9 \implies b = 3$ . The point is $(4 \sec \frac{\pi}{3}, 3 \tan \frac{\pi}{3}) = (4 \times 2, 3 \times \sqrt{3}) = (8, 3\sqrt{3})$ . The tangent equation is $\frac{x}{4} \sec \frac{\pi}{3} - \frac{y}{3} \tan \frac{\pi}{3} = 1$ . $\frac{x}{4} (2) - \frac{y}{3} (\sqrt{3}) = 1$ $\frac{x}{2} - \frac{\sqrt{3}y}{3} = 1$ $3x - 2\sqrt{3}y = 6$ .

What are the parametric equations for the hyperbola

\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

$x = b \tan \theta, y = a \sec \theta$

Learning Resources

Parametric Equations of a Hyperbola - Brilliant.org(documentation)

Provides a clear explanation of parametric equations for hyperbolas, including derivations and examples.

Conic Sections: Hyperbolas - Khan Academy(video)

A comprehensive video series covering hyperbolas, including their standard forms and properties, which can aid in understanding parametric forms.

Parametric Equations - Mathematics LibreTexts(documentation)

General introduction to parametric equations, useful for building foundational understanding before applying to specific conic sections.

Hyperbola Parametric Equations - YouTube Tutorial(video)

A visual tutorial demonstrating the derivation and application of parametric equations for hyperbolas.

Parametric Equations of Conics - University of British Columbia(documentation)

Explains parametric representations for various conic sections, including hyperbolas, with a focus on their geometric interpretation.

JEE Mathematics: Conic Sections - Hyperbola Notes(blog)

Provides detailed notes on hyperbolas, often including sections on parametric forms relevant to competitive exams.

Parametric Equations of Hyperbola - Toppr(documentation)

A concise guide specifically on the parametric equations of a hyperbola, with examples tailored for exam preparation.

Hyperbola - Wikipedia(wikipedia)

Offers a broad overview of hyperbolas, including their mathematical properties and various forms of equations, which can provide context.

Calculus: Parametric Equations and Polar Coordinates - Paul's Online Math Notes(documentation)

A comprehensive resource on parametric equations in calculus, with sections that can be applied to conic sections.

Problems on Parametric Equations of Hyperbola - Vedantu(blog)

Features solved problems and practice questions related to parametric equations of hyperbolas, ideal for exam practice.