Ellipse
Sum of distances to two foci is constant.
Key idea
Sum of distances to two foci is constant.
In depth
An ellipse is all points whose distances to two foci add to a constant. Standard form x²/a² + y²/b² = 1, with foci at distance c from the centre where c² = a² − b² when a > b.
Worked example
x²/25 + y²/9 = 1: a = 5, b = 3, c = 4, foci at (±4, 0).
Formula
Real-world uses
- Planetary orbits, whispering galleries and elliptical gears
- The shape of a running track’s centre line
Common mistake
Using c² = a² + b², which is for hyperbolas.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Ellipse?
Sum of distances to two foci is constant. It is a topic at the expert level in Geometry.
What is the formula for Ellipse?
The key formula is x²/a² + y²/b² = 1.
Where is Ellipse used in real life?
Real-world examples: Planetary orbits, whispering galleries and elliptical gears; The shape of a running track’s centre line.
What should I learn before Ellipse?
Make sure you are comfortable with Parabola (Geometric Definition), the previous topic in Geometry.
What is a common mistake with Ellipse?
Using c² = a² + b², which is for hyperbolas.
How can I practise Ellipse?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Ellipse?
Move on to Hyperbola next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Hyperbola – Difference of distances to two foci is constant.