How to Calculate Hyperbola Eccentricity: Formula & 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 What is the Eccentricity of a Hyperbola?
1. What is the formula for the eccentricity of a hyperbola?
The eccentricity of a hyperbola is given by the formula: e = \( \frac{\sqrt{a^2 + b^2}}{a} \) where a is the semi-transverse axis and b is the semi-conjugate axis. Here, \(e \gt 1\) for every hyperbola.
2. What is the formula of eccentricity?
The general formula for eccentricity of a conic section is: e = \( \frac{c}{a} \), where c is the distance from the center to the focus and a is the distance from the center to the vertex. For a hyperbola, c = \sqrt{a^2 + b^2}.
3. Is the eccentricity of a hyperbola less than 1?
No, the eccentricity of a hyperbola is always greater than 1. This is because, by definition, e = \frac{c}{a} \gt 1 for hyperbolas.
4. Can eccentricity be greater than 1?
Yes, the eccentricity is greater than 1 for a hyperbola. Ellipses have eccentricity between 0 and 1, parabolas have eccentricity equal to 1, and hyperbolas have eccentricity greater than 1.
5. What is the relation between eccentricity of parabola, ellipse, and hyperbola?
The eccentricity of different conics is:
• Ellipse: 0 < e < 1
• Parabola: e = 1
• Hyperbola: e > 1
6. What is the eccentricity of a circle?
The eccentricity of a circle is 0, because both foci are at the center, so the distance to the focus is zero.
7. What are 'a' and 'b' in the formula for hyperbola eccentricity?
In the equation of a hyperbola, 'a' represents the length of the semi-transverse axis (along the real axis), and 'b' is the length of the semi-conjugate axis (along the imaginary axis). They appear in the formula for eccentricity: e = \( \frac{\sqrt{a^2 + b^2}}{a} \).
8. What happens to the eccentricity of a hyperbola if b is greater than a?
If b > a, the value of \( \sqrt{a^2 + b^2} \) increases, so the eccentricity (e) also increases. This means the branches of the hyperbola become more ‘open’.
9. What is the eccentricity of a conjugate hyperbola?
The conjugate hyperbola has the same 'a' and 'b' as the original hyperbola, so it has the same eccentricity. Its equation differs by a swapped sign between the terms, but the value of e is unchanged.
10. How do you find the eccentricity of a hyperbola from its equation?
To find the eccentricity from the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \):
1. Identify a and b.
2. Use the formula e = \( \frac{\sqrt{a^2 + b^2}}{a} \).
11. What is the meaning of eccentricity of a hyperbola?
The eccentricity of a hyperbola is a measure of how 'stretched out' or 'open' the hyperbola is. A higher eccentricity means the branches are more widely separated. It is always greater than 1 for hyperbolas.
12. What is the value of eccentricity for a hyperbola whose asymptotes are perpendicular?
If the asymptotes of a hyperbola are perpendicular, the equation is \( x^2 - y^2 = a^2 \) (or b = a). Thus, e = \frac{\sqrt{2}a}{a} = \sqrt{2} (i.e., the eccentricity is \( \sqrt{2} \)).
