Vertices of an eclipse

High School

Definition

The vertices on a eclipse refers to the points of maximum curvature. On a eclipse there will always be two vertices directly across one another, and is always aligned with the line of symmetry as well as the X or Y axis

Worked examples

\(\frac{x^2}{25} + \frac{y^2}{9} = 1\)

This ellipse has vertices at \((5, 0)\) and \((-5, 0)\) because the larger denominator is under \(x^2\).

\(\frac{x^2}{4} + \frac{y^2}{16} = 1\)

Here the vertices are at \((0, 4)\) and \((0, -4)\) since the larger denominator is under \(y^2\).

Common mistakes

\(\frac{x^2}{9} + \frac{y^2}{25} = 1\) has vertices at \((3, 0)\) and \((-3, 0)\) → vertices are at \((0, 5)\) and \((0, -5)\) Vertices lie on the axis with the larger denominator; here 25 is under \(y^2\), so vertices are vertical.
confusing vertices with co-vertices (the endpoints on the minor axis) → vertices are the endpoints of the major axis only Vertices mark maximum curvature; co-vertices are on the shorter axis.
thinking an ellipse has four vertices → an ellipse has exactly two vertices The other two endpoints are co-vertices, not vertices.

Where you'll use it next

You'll use vertices when graphing and analyzing ellipses in precalculus, finding eccentricity and foci, and later in conic sections, physics orbits, and engineering applications.

Found in 1 StudyPug lesson

Conics - Ellipse

Grade 12 Math

12th Grade12th