What is Eccentricity of an Ellipse? Formula, Meaning & Calculation
The concept of ellipse plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding ellipses helps you solve problems in coordinate geometry, astronomy, engineering, and more!
What Is Ellipse?
An ellipse is a special curved shape that looks like a stretched circle or oval. In mathematics, an ellipse is defined as the set of all points in a plane such that the sum of their distances from two fixed points (called foci) is constant. You’ll find this concept applied in areas such as coordinate geometry, planetary motion, and design engineering.
Key Formula for Ellipse
Here’s the standard equation of an ellipse centered at the origin:
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
where a = length of the semi-major axis, and b = length of the semi-minor axis. If a > b, then the ellipse is longer along the x-axis. The distance between the center and each focus is \(c\), where \(c = \sqrt{a^2 - b^2}\).
| Parameter | Formula |
|---|---|
| Area | \(\pi ab\) |
| Eccentricity (e) | \(\sqrt{1-\dfrac{b^2}{a^2}}\) |
| Perimeter (approx.) | \(2\pi \sqrt{\dfrac{a^2+b^2}{2}}\) |
| Latus Rectum (L) | \(\dfrac{2b^2}{a}\) |
Parts of an Ellipse
- Center: The midpoint of the ellipse (usually at (0,0)).
- Foci: Two fixed points inside the ellipse (always along the major axis).
- Major Axis: The longest diameter passing through both foci.
- Minor Axis: The shortest diameter, perpendicular to the major axis.
- Vertices: Points where the ellipse crosses its axes.
- Eccentricity: A number showing how stretched the ellipse is. For circles, \(e=0\); for most ellipses, \(0<e<1\).
Cross-Disciplinary Usage
Ellipse is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions, such as planetary orbits, optics, engineering drafting, and even computer graphics for drawing curved paths.
Step-by-Step Illustration
- Find the area of an ellipse where \(a = 7\) cm and \(b = 5\) cm.
Area formula: \( \pi ab \)
- Insert values:
Area = \( \pi \times 7 \times 5 = 35\pi \approx 110 \) cm² (using \( \pi = 22/7 \) for exams)
Speed Trick or Vedic Shortcut
Here’s a quick tip for finding the eccentricity in MCQs:
- If you know \(a\) and \(b\), directly use e = \(\sqrt{1 - (b^2 / a^2)}\).
- If it’s a near-circle ellipse (like \(a = 6, b = 5.9\)), mental calculation shows \(e \approx 0\) (close to a circle).
- For exam speed, remember: “Smaller the difference between a and b, more circular is the ellipse.”
Tricks like these are practical in exams like boards, NTSE and JEE. Vedantu’s classes explain shortcuts with live problem solving for ellipses and all conic sections.
Try These Yourself
- Write the equation of an ellipse with major axis = 10 units, minor axis = 6 units, center at the origin.
- Calculate the eccentricity if \(a = 5, b = 3\).
- Is every circle an ellipse? Why?
- Sketch an ellipse on graph paper. Label all axes and foci.
Frequent Errors and Misunderstandings
- Mixing up which is major and minor axis (a always refers to the longer one).
- Using the area formula for a circle on an ellipse (πab vs πr2).
- Forgetting that eccentricity for ellipse is always 0 < e < 1 (not equal to or more than 1).
- Incorrectly plotting or labeling foci and vertices on diagrams.
Relation to Other Concepts
The idea of ellipse connects closely with topics such as circle equations, parabola, and hyperbola. Mastering ellipses helps with coordinate geometry and solving different types of conic section problems. You may also want to understand eccentricity to see how conic sections change shape.
Classroom Tip
A quick way to remember the ellipse equation is: “Sum of the distances from any point on the curve to the two foci = constant (major axis length)”. Draw a string ‘loop’ pinned at two foci—the traced path forms an ellipse! Vedantu’s teachers often use real-life orbits and string models to help visualize ellipses.
Wrapping It All Up
We explored ellipse—from its definition, formula, worked example, quick tricks, mistakes to avoid, and connections to other Maths concepts. Keep practicing with more problems and use live interactive sessions at Vedantu to clear your doubts and master the topic. The ellipse isn’t just an exam question—it’s a key shape in science, engineering, and design!
Explore Related Topics
- Area of Ellipse (Formulas & Examples)
- Eccentricity (Ellipse, Circle, Hyperbola)
- Circle Equation and Properties
FAQs on Ellipse in Maths – Definition, Equation, Properties & Examples
1. What is an ellipse in mathematics?
An ellipse is a closed, oval-shaped curve defined as the set of all points in a plane such that the sum of the distances to two fixed points (called foci) is constant. It's a type of conic section, formed by the intersection of a plane and a cone. A special case of an ellipse is a circle, where the two foci coincide.
2. What is the standard equation of an ellipse?
The standard equation of an ellipse depends on its orientation. If the major axis is horizontal and the center is at the origin (0,0), the equation is: x²/a² + y²/b² = 1, where 'a' is the length of the semi-major axis (half the longest diameter) and 'b' is the length of the semi-minor axis (half the shortest diameter). If the major axis is vertical, the equation is: y²/a² + x²/b² = 1.
3. How do I find the area of an ellipse?
The area of an ellipse is given by the formula: Area = πab, where 'a' and 'b' are the lengths of the semi-major and semi-minor axes, respectively.
4. What is the eccentricity of an ellipse and how is it calculated?
Eccentricity (e) is a measure of how elongated an ellipse is. It's calculated as the ratio of the distance from the center to a focus (c) to the length of the semi-major axis (a): e = c/a. The eccentricity of an ellipse always lies between 0 and 1 (0 < e < 1). An eccentricity of 0 represents a circle.
5. What are the major and minor axes of an ellipse?
The major axis is the longest diameter of the ellipse, passing through the center and both foci. The minor axis is the shortest diameter of the ellipse, passing through the center and perpendicular to the major axis.
6. What are the foci of an ellipse?
The foci (plural of focus) are two fixed points inside the ellipse. The sum of the distances from any point on the ellipse to the two foci is constant.
7. How can I graph an ellipse?
To graph an ellipse, you need its equation. Identify the center, semi-major axis (a), and semi-minor axis (b). Plot the center. From the center, measure 'a' units horizontally (or vertically, depending on the equation) to find the vertices. Measure 'b' units vertically (or horizontally) to find the co-vertices. Sketch the ellipse through these points.
8. What are some real-life applications of ellipses?
Ellipses appear in many real-world contexts, including: planetary orbits (Kepler's Laws), whispering galleries (sound reflection), engineering designs (bridges, arches), and various optical systems (lenses, telescopes).
9. What is the difference between an ellipse and a circle?
A circle is a special case of an ellipse where both foci coincide at the center, resulting in a perfectly round shape (eccentricity = 0). An ellipse has two distinct foci, leading to an oval shape (0 < eccentricity < 1).
10. How is the latus rectum related to an ellipse?
The latus rectum of an ellipse is a chord perpendicular to the major axis and passing through a focus. Its length is given by 2b²/a, where 'a' and 'b' are the semi-major and semi-minor axes respectively.
11. What is the relationship between the semi-major axis, semi-minor axis, and distance to the foci (c)?
The relationship is expressed by the equation: a² = b² + c², where 'a' is the length of the semi-major axis, 'b' is the length of the semi-minor axis, and 'c' is the distance from the center to a focus.
12. Can you explain the concept of a directrix in relation to an ellipse?
A directrix is a line associated with an ellipse. For every point on the ellipse, the ratio of its distance to a focus and its distance to the corresponding directrix is constant and equal to the eccentricity (e).
