Hyperbola

Intuition and core definition

A hyperbola is the set of points for which the absolute difference of distances to two foci is constant. Standard forms have one positive and one negative squared term; the positive term identifies the direction the two branches open.

Notation, language, and conditions

Horizontal form is $(x-h)^2/a^2-(y-k)^2/b^2=1$; vertical form reverses the terms. The centre is $(h,k)$, vertices lie $a$ units along the transverse axis, foci lie $c$ units with $c^2=a^2+b^2$, and asymptotes guide branch behaviour.

Why this idea matters

A hyperbola consists of two branches governed by a constant difference of focal distances and approached by asymptote lines.

A dependable method

1. Rewrite the equation in standard form equal to $1$.
2. Read the centre and identify the positive squared term.
3. Take square roots to obtain $a$ and $b$.
4. Find vertices and compute foci using $c^2=a^2+b^2$.
5. Draw the guiding rectangle and asymptotes, then sketch branches through vertices.

Worked example

Common mistakes