Major Axis of a Hyperbola

High School

Definition

The principle axis of symmetry in a hyperbola that seperates it into two mirrored halves. It is a line that passes through the vertices, center, and the foci of a hyperbola that ends at the widest points of a hyperbola's perimeter.

Worked examples

\(\frac{x^2}{9} - \frac{y^2}{4} = 1\)

Horizontal hyperbola; the major axis lies along the x-axis from \((-3,0)\) to \((3,0)\) through the vertices.

\(\frac{y^2}{16} - \frac{x^2}{9} = 1\)

Vertical hyperbola; the major axis lies along the y-axis from \((0,-4)\) to \((0,4)\) through the vertices.

Common mistakes

The major axis connects the two branches at their closest points. → The major axis passes through the vertices, which are the closest points on each branch. The vertices lie on the major axis; it doesn't 'connect' the branches but passes through their tips.
In \(\frac{x^2}{9} - \frac{y^2}{16} = 1\), the major axis has length 8. → The major axis has length \(2a = 6\); here \(a^2 = 9\) so \(a = 3\). Use the denominator under the positive term: \(a^2 = 9\), not the larger denominator.
The foci lie on the minor axis. → The foci lie on the major axis, beyond the vertices. The major axis passes through center, vertices, and foci; the minor axis is perpendicular.

Where you'll use it next

You'll use the major axis when graphing hyperbolas, finding vertices and foci, writing equations in standard form, and analyzing conic sections in precalculus and calculus.

Found in 1 StudyPug lesson

Conics - Hyperbola: Mastering the Art of Curved Geometry

Grade 12 Math

Dive into the world of hyperbolas! Discover key components, graphing techniques, and real-world applications. Perfect for students seeking to excel in conic sections and analytical geometry.

12th Grade12th