Hyperbola
Difference of distances to two foci is constant.
Key idea
Difference of distances to two foci is constant.
In depth
A hyperbola is all points whose distances to two foci differ by a constant. Standard form x²/a² − y²/b² = 1, with asymptotes y = ±(b/a)x and c² = a² + b².
Worked example
x²/16 − y²/9 = 1: asymptotes y = ±(3/4)x, foci (±5, 0).
Formula
Real-world uses
- Long-range navigation systems
- The shape of cooling towers and the paths of some comets
Common mistake
Confusing which term is positive, which sets the opening direction.
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 Hyperbola?
Difference of distances to two foci is constant. It is a topic at the expert level in Geometry.
What is the formula for Hyperbola?
The key formula is x²/a² − y²/b² = 1.
Where is Hyperbola used in real life?
Real-world examples: Long-range navigation systems; The shape of cooling towers and the paths of some comets.
What should I learn before Hyperbola?
Make sure you are comfortable with Ellipse, the previous topic in Geometry.
What is a common mistake with Hyperbola?
Confusing which term is positive, which sets the opening direction.
How can I practise Hyperbola?
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 Hyperbola?
Move on to Eccentricity of Conics 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.
- Ellipse – Sum of distances to two foci is constant.