The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Planetary orbits are ellipses (Kepler's first law).
Definition
The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.
💡 Intuition
Imagine pinning two ends of a loose string to a board (these are the foci), then tracing a curve with a pencil keeping the string taut. The resulting oval shape is an ellipse. A circle is just a special ellipse where both foci coincide.
🎯 Core Idea
An ellipse generalizes a circle by having two axes of different lengths. The relationship c^2 = a^2 - b^2 connects the foci to the axes. When a = b, the ellipse becomes a circle.
Example
Formula
Foci: c^2 = a^2 - b^2 (where a > b). Eccentricity: e = \frac{c}{a} (with 0 \leq e < 1).
Notation
a = semi-major axis (longer), b = semi-minor axis (shorter), c = distance from center to each focus.
🌟 Why It Matters
Planetary orbits are ellipses (Kepler's first law). Ellipses appear in optics (whispering galleries), satellite orbits, medical imaging (lithotripsy), and architecture.
💭 Hint When Stuck
Compare the two denominators. The larger one is a^2 and tells you which direction the major axis goes. Then find c using c^2 = a^2 - b^2.
Formal View
Related Concepts
🚧 Common Stuck Point
The larger denominator is always a^2, regardless of whether it's under x or y. If a^2 is under y, the major axis is vertical.
⚠️ Common Mistakes
- Confusing which denominator is a^2: a is ALWAYS the larger value. If the larger denominator is under y, the ellipse is taller than it is wide.
- Using c^2 = a^2 + b^2 (hyperbola formula) instead of c^2 = a^2 - b^2 (ellipse formula). For ellipses, c < a; for hyperbolas, c > a.
- Forgetting to take square roots: if a^2 = 25, then a = 5, not 25. The semi-axis lengths are a and b, not a^2 and b^2.
Go Deeper
Frequently Asked Questions
What is Ellipse in Math?
The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.
Why is Ellipse important?
Planetary orbits are ellipses (Kepler's first law). Ellipses appear in optics (whispering galleries), satellite orbits, medical imaging (lithotripsy), and architecture.
What do students usually get wrong about Ellipse?
The larger denominator is always a^2, regardless of whether it's under x or y. If a^2 is under y, the major axis is vertical.
What should I learn before Ellipse?
Before studying Ellipse, you should understand: equation of circle.
Prerequisites
Next Steps
Cross-Subject Connections
How Ellipse Connects to Other Ideas
To understand ellipse, you should first be comfortable with equation of circle. Once you have a solid grasp of ellipse, you can move on to conic sections overview and hyperbola.