Ellipse Standard Equations: Explained with Examples

An ellipse is the set of points for which the sum of distances to two fixed points (the foci) is constant. Its standard equation is x²/a² + y²/b² = 1 with a > b > 0: the major axis lies on the x-axis with vertices (±a, 0), the minor axis on the y-axis with vertices (0, ±b), and the foci at (±ae, 0) where the eccentricity e = sqrt(1 − b²/a²).

Standard forms matter because they make every feature readable: the bigger denominator names the major axis, c = ae gives the focal distance, and the latus rectum 2b²/a gives the width at the focus. Ellipse questions in exams are almost always "read the parameters and sketch" — and planetary orbits, the most famous ellipses of all, are described exactly this way.

Key Concepts

• Standard form (major axis x): x²/a² + y²/b² = 1, a > b; vertices (±a, 0), co-vertices (0, ±b), foci (±ae, 0).
• Standard form (major axis y): x²/b² + y²/a² = 1, a > b; vertices (0, ±a), foci (0, ±ae).
• Eccentricity: e = sqrt(1 − b²/a²), with 0 < e < 1; e = 0 is a circle, e near 1 is very flat.
• Key lengths: major axis 2a, minor axis 2b, focal distance 2ae, latus rectum 2b²/a.

Formulas

• x²/a² + y²/b² = 1 (a > b): centre (0, 0); foci (±ae, 0); directrices x = ±a/e; vertices (±a, 0).
• Eccentricity e = c/a where c² = a² − b², i.e. e = sqrt(1 − b²/a²).
• Latus rectum: length 2b²/a; endpoints (±ae, ±b²/a).
• Units: a, b, c carry length units; e is dimensionless.
• Assumption: a > b > 0; if the denominators are equal the curve is a circle (e = 0).

Worked Example

For the ellipse x²/25 + y²/16 = 1, find the vertices, foci, eccentricity and latus rectum.

1. Match the form: a² = 25 so a = 5; b² = 16 so b = 4. Since 25 > 16, the major axis is the x-axis.
2. Vertices: (±5, 0); co-vertices: (0, ±4).
3. Focal distance: c² = a² − b² = 25 − 16 = 9, so c = 3; foci (±3, 0).
4. Eccentricity: e = c/a = 3/5 = 0.6.
5. Latus rectum: length 2b²/a = 32/5 = 6.4; endpoints (3, ±16/5) and (−3, ±16/5).
6. Verify the definition at (5, 0): distances to foci are |5 − 3| = 2 and |5 + 3| = 8, summing to 10 = 2a. Correct.

Common Misconceptions

• "a is always under x²." a² is the LARGER denominator; if y² has the bigger one, the major axis is vertical.
• "e = b/a." Eccentricity is e = sqrt(1 − b²/a²) = c/a, not b/a.
• "The foci are at (±b, 0)." They are at (±ae, 0) — a distance c = sqrt(a² − b²) from the centre.

Real-World Applications

• Planetary orbits: planets travel on ellipses with the Sun at one focus (Kepler's first law).
• Whispering galleries: sound from one focus of an elliptical room reflects to the other focus.
• Medical lithotripsy: elliptical reflectors focus shock waves from one focus onto a kidney stone at the other.
• Engineering: elliptical gears and cams convert rotation with a varying ratio, designed from the standard equation.

In short: x²/a² + y²/b² = 1 with a > b puts the major axis on x — read a, b, compute c = sqrt(a² − b²), and every feature (foci, eccentricity, latus rectum) follows. Double-check which denominator is larger before naming the major axis, and verify with the constant-sum property at a vertex.