Mastering Hyperbolas: Standard Equations for Competitive Exams

Welcome to this module on the standard equations of a hyperbola, a crucial topic for competitive exams like JEE. A hyperbola is a fascinating conic section defined by a set of points where the difference of the distances to two fixed points (foci) is constant. Understanding its standard forms is key to solving a wide range of problems.

Defining the Hyperbola

A hyperbola is the locus of points in a plane such that the absolute difference of the distances from any point on the locus to two fixed points (the foci) is a constant. This constant difference is equal to 2a, where 'a' is the distance from the center to a vertex.

The hyperbola's shape is determined by its foci and the constant difference of distances.

Imagine two points, the foci. For any point on the hyperbola, moving towards one focus and then away from the other, the total path length difference will always be the same. This creates the characteristic two-branched, open curve.

Mathematically, if F1 and F2 are the foci and P is any point on the hyperbola, then $|PF_1 - PF_2| = 2a$ , where $2a$ is a positive constant. The distance between the foci, $2c$ , must be greater than $2a$ (i.e., $c > a$ ). The relationship between $a$ , $b$ , and $c$ for a hyperbola is $c^2 = a^2 + b^2$ , where $b$ is related to the conjugate axis.

Standard Equation Type 1: Transverse Axis along the x-axis

When the transverse axis (the axis containing the foci and vertices) lies along the x-axis and the center is at the origin (0,0), the standard equation of the hyperbola is:

The standard equation for a hyperbola centered at the origin with its transverse axis along the x-axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ . In this form, the vertices are located at $(\pm a, 0)$ . The foci are at $(\pm c, 0)$ , where $c^2 = a^2 + b^2$ . The asymptotes are the lines $y = \pm \frac{b}{a}x$ . The value of 'a' determines the distance from the center to the vertices along the transverse axis, and 'b' is related to the conjugate axis.

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

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

Key characteristics for this form:

Vertices: $(\pm a, 0)$
Foci: $(\pm c, 0)$ , where $c^2 = a^2 + b^2$
Asymptotes: $y = \pm \frac{b}{a}x$
The transverse axis is horizontal.

Standard Equation Type 2: Transverse Axis along the y-axis

Conversely, if the transverse axis lies along the y-axis and the center is at the origin (0,0), the standard equation of the hyperbola is:

Feature	Transverse Axis along x-axis	Transverse Axis along y-axis
Standard Equation	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
Vertices	$(\pm a, 0)$	$(0, \pm a)$
Foci	$(\pm c, 0)$	$(0, \pm c)$
Asymptotes	$y = \pm \frac{b}{a}x$	$y = \pm \frac{a}{b}x$
Relationship $c^2$	$a^2 + b^2$	$a^2 + b^2$

For a hyperbola

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

, where are the vertices located?

The vertices are located at $(0, \pm a)$ .

Remember: The term with the positive coefficient in the standard equation indicates the direction of the transverse axis. If $x^2$ is positive, the transverse axis is horizontal; if $y^2$ is positive, it's vertical.

Hyperbolas with Center at (h, k)

When the center of the hyperbola is shifted to a point $(h, k)$ , we replace $x$ with $(x-h)$ and $y$ with $(y-k)$ in the standard equations. This is a direct application of translation rules.

For a hyperbola with center $(h, k)$ and transverse axis parallel to the x-axis: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ Vertices: $(h \pm a, k)$ Foci: $(h \pm c, k)$

For a hyperbola with center $(h, k)$ and transverse axis parallel to the y-axis: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ Vertices: $(h, k \pm a)$ Foci: $(h, k \pm c)$

If a hyperbola has its center at (2, 3) and its transverse axis is horizontal, what is the form of its equation?

$\frac{(x-2)^2}{a^2} - \frac{(y-3)^2}{b^2} = 1$

Key Takeaways for Problem Solving

To solve problems involving hyperbolas:

Identify the Center: Look for $(h, k)$ in the equation.
Determine the Transverse Axis: Check which squared term is positive.
Find 'a' and 'b': These are the square roots of the denominators of the positive and negative terms, respectively.
Calculate 'c': Use $c^2 = a^2 + b^2$ .
Locate Vertices and Foci: Use the center and the values of $a$ and $c$ based on the transverse axis.
Identify Asymptotes: Use the formulas $y-k = \pm \frac{b}{a}(x-h)$ or $y-k = \pm \frac{a}{b}(x-h)$ .

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Learning Resources

Hyperbolas - Definition, Equation, Properties & Examples(documentation)

Provides a concise definition, standard equations, and key properties of hyperbolas, useful for quick reference.

Conic Sections: Hyperbolas - Varsity Tutors(documentation)

Explains the definition, standard forms, and key components of a hyperbola with clear diagrams and examples.

Hyperbola - Wikipedia(wikipedia)

A comprehensive overview of the hyperbola, including its mathematical definition, history, and various applications.

Hyperbola Standard Equation | Maths(documentation)

A beginner-friendly explanation of the hyperbola's standard equation, properties, and how to graph it.

Hyperbola - Standard Form - YouTube(video)

A video tutorial demonstrating how to derive and use the standard form of a hyperbola's equation.

Conic Sections - Hyperbolas - Khan Academy(video)

An introductory video to hyperbolas, covering their definition and basic properties.

JEE Mathematics: Conic Sections - Hyperbola(video)

A JEE-focused video explaining the properties and standard equations of hyperbolas with solved examples.

Hyperbola - Equation, Properties, Graph, Examples(documentation)

Details the standard equations, properties, and graphical representation of hyperbolas with illustrative examples.

Conic Sections - Hyperbolas - Brilliant.org(documentation)

Explores the definition, standard equations, and key concepts of hyperbolas with interactive elements.

Properties of Hyperbola - Toppr(documentation)

Summarizes the essential properties of a hyperbola, including its standard equations and related terms.

Standard Equations of a Hyperbola