Ellipse

High School

Definition

A shape that looks like a stretched circle, giving it the appearance of an oval (keep in mind though that in mathematics, an ellipse is not an oval). It is a curve that is surrounded by two focal points of which the sum of the distances of the two foci are constant no matter which point you are at on the curve.

Worked examples

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

An ellipse centered at the origin with horizontal semi-major axis 5 and vertical semi-minor axis 3.

\(\frac{(x-2)^2}{16} + \frac{(y+1)^2}{25} = 1\)

Center at \((2, -1)\); vertical major axis of length 10 and horizontal minor axis of length 8.

\(c^2 = a^2 - b^2 \)→\( c^2 = 25 - 9 = 16 \)→\( c = 4\)

Finding the distance from center to each focus when the semi-major axis is 5 and semi-minor axis is 3.

Common mistakes

\(\frac{x^2}{25} + \frac{y^2}{9} = 1\) has foci at \((\pm 5, 0)\) → \(c = \sqrt{25 - 9} = 4\), so foci are at \((\pm 4, 0)\) The foci are c units from center, where \(c^2 = a^2 - b^2\), not at the vertices.
Thinking the larger denominator is always under \(x^2\) → The larger denominator determines the major axis direction If the larger denominator is under \(y^2\), the ellipse is taller than it is wide.
Calling every ellipse an oval → An ellipse has a precise definition involving two foci In mathematics, 'oval' is informal; an ellipse is defined by the constant sum of distances to two foci.

Where you'll use it next

You'll graph and analyze ellipses in precalculus and analytic geometry, solve systems involving conics in algebra, and encounter ellipses in physics (planetary orbits) and engineering (reflective properties).

Found in 1 StudyPug lesson

Conics - Ellipse

Grade 12 Math

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