Foci Vertices and Latus Rectum of an Ellipse: Notes and FAQs

The foci of an ellipse are the two fixed points that define it: the sum of the distances from any point on the ellipse to the two foci is constant, equal to 2a. The vertices are the points where the ellipse crosses its major and minor axes, and the latus rectum is the chord through a focus perpendicular to the major axis. For x²/a² + y²/b² = 1 with a > b, the foci are at (±ae, 0), the vertices at (±a, 0) and (0, ±b), and each latus rectum has length 2b²/a.

These are the first features you mark when sketching an ellipse, and exams ask for them constantly. The foci give the ellipse its special reflecting property — rays from one focus bounce to the other — while the latus rectum fixes the ellipse's width at its focal points.

Key Concepts

• Focus (plural foci): a fixed point of the ellipse; for a > b the foci are at (±c, 0) with c = ae, or at (0, ±c) if the major axis is vertical.
• Vertex: an endpoint of the major axis, (±a, 0) in the horizontal case.
• Co-vertex: an endpoint of the minor axis, (0, ±b) in the horizontal case.
• Major axis (length 2a) passes through both foci; the minor axis (length 2b) is perpendicular to it.
• Latus rectum: the chord through a focus perpendicular to the major axis; length 2b²/a; there are two of them.

Formulas

• Foci: (±ae, 0) for a horizontal major axis; (0, ±ae) for a vertical one.
• Vertices (±a, 0); co-vertices (0, ±b).
• Latus rectum: L = 2b²/a, with endpoints (±c, ±b²/a) in the horizontal case.
• Units: a, b, c, L are lengths; e is dimensionless. Assumes standard position with centre at the origin.

Worked Example

Find the foci, vertices and latus rectum of 16x² + 9y² = 144.

1. Standard form. Divide by 144: x²/9 + y²/16 = 1. The larger denominator is under y², so the major axis is vertical: a² = 16 (a = 4), b² = 9 (b = 3).
2. Find c. c² = 16 − 9 = 7, so c = √7 ≈ 2.65.
3. Foci. (0, ±√7); eccentricity e = √7/4 ≈ 0.66.
4. Vertices and co-vertices. Vertices: (0, ±4); co-vertices: (±3, 0).
5. Latus rectum. 2·9/4 = 4.5.

Answer: foci (0, ±√7), vertices (0, ±4), co-vertices (±3, 0), latus rectum 4.5.

Common Misconceptions

• "The foci always lie on the x-axis." No — they lie on the major axis, which is the axis with the larger denominator; here it was vertical.
• "All four axis-crossings are vertices." Only the major-axis ends are vertices; the minor-axis ends are co-vertices.
• "In 2b²/a, b means the term under x²." No — a is always the semi-major axis (the larger one), whichever letter sits under x².

Real-World Applications

• Satellite dishes: an elliptical dish concentrates incoming signals at the focus where the receiver sits.
• Medical lithotripsy: shock waves generated at one focus of an elliptical reflector are focused onto a kidney stone at the other focus.
• Planetary motion: the Sun sits at one focus of each planet's elliptical orbit.
• Drawing ellipses: two pegs at the foci and a string loop of length 2a (the gardener's method) traces a perfect ellipse.

To recap: foci sit on the major axis at distance ae from the centre, vertices and co-vertices are the axis-crossings, and the latus rectum 2b²/a gives the width at each focus. Identify the larger denominator first.