What Is the Eccentricity of a Hyperbola Formula Derivation and Solved Examples
Eccentricity of Hyperbola is an important concept in conic sections, especially in board exams and competitive entrance tests. It helps you measure how stretched or circular a hyperbola is, links geometry with algebra, and appears in both objective questions and problem solving. Understanding it can boost your confidence for topics connected to ellipses, circles, and parabola.
Formula Used in Eccentricity of Hyperbola
The standard formula is: \( e = \sqrt{1 + \dfrac{b^2}{a^2}} \), where e is the eccentricity, a is the length of the semi-transverse axis, and b is the length of the semi-conjugate axis of the hyperbola.
Here’s a helpful table to understand eccentricity of hyperbola more clearly:
Eccentricity of Hyperbola Table
| Conic | Standard Eccentricity Formula | Value of e |
|---|---|---|
| Circle | e = 0 | 0 |
| Ellipse | e = √(1 - b2/a2) | 0 < e < 1 |
| Hyperbola | e = √(1 + b2/a2) | e > 1 |
| Parabola | e = 1 | 1 |
This table shows how the pattern of eccentricity of hyperbola compares with other conic sections like ellipses and circles. For more details, see our comparison page: Difference between Parabola and Hyperbola.
Worked Example – Solving a Problem
Let’s solve a common exam question for eccentricity of hyperbola:
1. Given the equation: \( \frac{x^2}{25} - \frac{y^2}{9} = 1 \) 2. Identify a2 and b2 from the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \):
b2 = 9 ⇒ b = 3
3. Write the formula: \( e = \sqrt{1 + \frac{b^2}{a^2}} \) 4. Substitute the values:
5. Calculate the value:
6. Therefore, the eccentricity of this hyperbola is approximately 1.17.
For more solved examples, check Hyperbola Important Questions.
Practice Problems
- Find the eccentricity of the hyperbola \( \frac{x^2}{100} - \frac{y^2}{36} = 1 \).
- If the eccentricity of a hyperbola is 1.25 and a = 8, find b.
- Is it possible for a hyperbola to have eccentricity less than 1? Why or why not?
- Compare the eccentricity of the hyperbola with that of an ellipse having the same value of a.
Common Mistakes to Avoid
- Confusing eccentricity of hyperbola formula with that of ellipse (Ellipse page explains this clearly).
- Using wrong values for a and b from the equation arrangement. Always check standard forms.
- Assuming the eccentricity can be less than 1 for a hyperbola. Remember, for hyperbolas, e > 1.
- Not squaring a and b when substituting in formula.
Real-World Applications
The concept of eccentricity of hyperbola is used in satellite dish designs, optics, navigation systems, and certain architectural curves. Engineers use it when analyzing reflective properties and orbits. Vedantu helps students relate these formulas to real problems seen in exams and technology fields. Explore conic links at Conic Sections.
We explored the idea of eccentricity of hyperbola, learned its formula, solved exam-type problems, and saw why its value is always above one. For more practice and deeper learning, visit Vedantu’s resources to connect hyperbola and related conic topics with real life and entrance exams.
Related links:
Eccentricity of Ellipse |
Parametric Form of Hyperbola |
Conic Sections |
Difference between Parabola and Hyperbola
FAQs on Eccentricity of Hyperbola Explained with Formula
1. What is the eccentricity of a hyperbola?
The eccentricity of a hyperbola is a measure of how much the hyperbola deviates from a circle, and it is always greater than 1. It is defined as the ratio of the distance of any point on the hyperbola from a focus to its perpendicular distance from the corresponding directrix. For every hyperbola, e > 1, which distinguishes it from an ellipse (where e < 1) and a parabola (where e = 1).
2. What is the formula for the eccentricity of a hyperbola?
The formula for the eccentricity (e) of a hyperbola is e = c/a, where c is the distance from the center to a focus and a is the distance from the center to a vertex. For the standard form hyperbola x²/a² − y²/b² = 1:
- c² = a² + b²
- e = c/a = √(a² + b²)/a
3. How do you find the eccentricity of a hyperbola from its equation?
To find the eccentricity of a hyperbola, first write the equation in standard form and then use e = c/a. Steps:
- Write the equation as x²/a² − y²/b² = 1 or y²/a² − x²/b² = 1.
- Identify a² and b².
- Compute c² = a² + b².
- Find e = c/a.
4. Why is the eccentricity of a hyperbola always greater than 1?
The eccentricity of a hyperbola is always greater than 1 because c² = a² + b², which makes c greater than a. Since e = c/a and c > a, the ratio must be greater than 1. This property geometrically explains why a hyperbola opens outward more widely than a parabola.
5. What is the eccentricity of the hyperbola x²/4 − y²/9 = 1?
The eccentricity of the hyperbola x²/4 − y²/9 = 1 is √13/2. Solution steps:
- a² = 4, so a = 2
- b² = 9
- c² = a² + b² = 4 + 9 = 13
- c = √13
- e = c/a = √13/2
6. What is the relationship between a, b, c and eccentricity in a hyperbola?
In a hyperbola, the relationship between a, b, c and eccentricity is given by c² = a² + b² and e = c/a. Here:
- a = distance from center to vertex
- b = related to the conjugate axis
- c = distance from center to focus
7. How does eccentricity affect the shape of a hyperbola?
The eccentricity determines how wide or narrow a hyperbola opens. If e is just slightly greater than 1, the branches are relatively narrow. If e is much greater than 1, the branches open more widely. Since e = c/a, increasing c (while keeping a fixed) makes the hyperbola more spread out.
8. What is the eccentricity of a rectangular hyperbola?
The eccentricity of a rectangular hyperbola is √2. In a rectangular hyperbola, a = b, so:
- c² = a² + a² = 2a²
- c = a√2
- e = c/a = √2
9. What is the difference between the eccentricity of a hyperbola and an ellipse?
The key difference is that the eccentricity of a hyperbola is greater than 1, while the eccentricity of an ellipse lies between 0 and 1. For an ellipse, c² = a² − b² and e = c/a < 1. For a hyperbola, c² = a² + b² and e = c/a > 1. This difference explains why ellipses are closed curves and hyperbolas are open curves.
10. Can the eccentricity of a hyperbola be equal to 1?
The eccentricity of a hyperbola can never be equal to 1 because by definition e = c/a and c is always greater than a. If e were equal to 1, the conic section would be a parabola, not a hyperbola. Therefore, for every hyperbola, e > 1.
