Functions
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The Hyperbola

Asymptotes, foci, and why the curve never touches

A hyperbola looks like two mirror-image curves opening in opposite directions. Its equation is \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 — notice the minus sign, which is what makes it different from an ellipse (which has a plus sign).

Every hyperbola has asymptotes — straight lines that the curve gets infinitely close to but never touches. It also has two foci, and a defining property: the absolute difference of distances from any point on the curve to the two foci is constant.

In this lesson, you'll use sliders for a and b to reshape the hyperbola, see how the asymptotes change, locate the foci, and discover applications from GPS satellites to the shape of shadows.

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FAQ

What is a hyperbola?
A hyperbola is a conic section with two separate branches. The standard form \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 opens left and right. The minus sign is what distinguishes it from an ellipse. The distance between the two vertices is 2a.
What are asymptotes of a hyperbola?
The asymptotes are the lines y = \pm \frac{b}{a} x. The hyperbola gets infinitely close to these lines as x → ±∞ but never actually touches them. They form an "X" shape that guides the curve's direction.
How do I find the foci of a hyperbola?
For a hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, the foci are at (\pm c, 0) where c = \sqrt{a^2 + b^2}. Note the plus sign — for an ellipse it's minus, but for a hyperbola it's plus, so the foci are always outside the vertices.
What is the constant-difference property?
For any point P on the hyperbola, |d(P, F_1) - d(P, F_2)| = 2a. The absolute difference of distances to the two foci is always 2a. This contrasts with the ellipse, where the sum (not difference) is constant.

Related Topics

Quadratic EquationsThe Unit CirclePythagorean Theorem