Definition: A hyperbola is the set of points whose distances to two foci differ by a constant; it has two separate branches that approach slanted asymptotes.
What defines a hyperbola?
The difference of the distances to the two foci is constant. A hyperbola opening left and right, centered at the origin, has equation x2/a2 – y2/b2 = 1. The foci satisfy c2 = a2 + b2, and the asymptotes are y = ±(b/a)x.
How do you sketch it?
Worked example: x2/16 – y2/9 = 1. Here a = 4 and b = 3.
- Vertices: (±4, 0)
- c = square root of (16 + 9) = 5, so the foci are (±5, 0)
- Asymptotes: y = ±(3/4)x
Draw the box with corners (±4, ±3), extend its diagonals, and bend each branch toward them.
What is a typical error?
Deciding the opening direction by the larger denominator. Instead, look at which squared term is positive: if x2 comes first with a plus sign, the branches open left and right. Mathbot checks the sign, not the size.
Example: x^2/16 – y^2/9 = 1 has foci (±5, 0) and asymptotes y = ±(3/4)x