AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.6B — Conic Sections — Ellipses
Notes — Two Foci and a Sum of Distances
💡 Learning Objectives (4.6.A Part 2)
By the end of this lesson you will be able to:
- Define an ellipse in terms of two foci and a constant sum of distances
- Identify center, vertices, foci, and major/minor axes from the equation
- Write the standard form equation of an ellipse from given features
- Sketch an ellipse and find its eccentricity
1. Definition of an Ellipse
An ellipse is the set of all points whose distances to two fixed points (the FOCI) sum to a constant.
Pin two ends of a string to a board (the foci); pull the string taut with a pencil; trace. The pencil draws an ellipse, because the string length (the sum of distances) is constant. This sum equals 2a, where a is the SEMI-MAJOR AXIS.
2. Standard Forms
💡 Standard Forms of an Ellipse Centered at (h, k)
- Horizontal major axis (a > b): ((x − h)²)/a² + ((y − k)²)/b² = 1
- Vertical major axis (a > b): ((x − h)²)/b² + ((y − k)²)/a² = 1
- a = semi-major axis (largest radius); b = semi-minor axis
- Foci on the major axis at distance c from center, where c² = a² − b²
⚠️ How to tell horizontal from vertical major axis
Look at which denominator is LARGER. If the denominator under (x − h)² is larger, the major axis is horizontal. If the denominator under (y − k)² is larger, it's vertical.
3. Worked Example
📘 Example — Identify features of (x − 2)²/25 + (y + 1)²/9 = 1
- Center: (2, −1)
- a² = 25, so a = 5; b² = 9, so b = 3 (denominator under x is larger ⇒ horizontal major axis)
- Vertices (along the major axis): (2 ± 5, −1) = (7, −1) and (−3, −1)
- Co-vertices (along minor axis): (2, −1 ± 3) = (2, 2) and (2, −4)
- c² = 25 − 9 = 16, so c = 4. Foci at (2 ± 4, −1) = (6, −1) and (−2, −1)
4. Writing the Equation from Features
📘 Example — Write the equation
Ellipse with vertices (1, 5) and (1, −3), and foci (1, 3) and (1, −1).
- Center: midpoint of vertices = (1, 1)
- Distance vertex-to-center: a = 4 (vertical, since vertices share x)
- Distance focus-to-center: c = 2
- b² = a² − c² = 16 − 4 = 12
- Vertical major axis: (x − 1)²/12 + (y − 1)²/16 = 1
5. Eccentricity
The ECCENTRICITY of an ellipse is e = c/a, where 0 ≤ e < 1. It measures how ‘stretched’ the ellipse is:
- e = 0: circle (foci coincide at center; a = b)
- e close to 0: nearly circular ellipse
- e close to 1: very elongated ellipse (foci near vertices)
6. Sketching an Ellipse
- Plot the center
- From the center, mark a units along the major axis to find the vertices
- Mark b units along the minor axis to find the co-vertices
- Mark c units along the major axis to find the foci
- Connect the four ‘tip’ points with a smooth oval
7. Reflective Property
A ray from one focus reflects off the ellipse and passes through the OTHER focus. This is the principle behind ‘whispering galleries’ — a sound from one focus is heard clearly at the other.
8. Converting from General Form
📘 Example — Convert 4x² + 9y² − 16x + 18y − 11 = 0
- Group: 4(x² − 4x) + 9(y² + 2y) = 11
- Complete the square: 4((x − 2)² − 4) + 9((y + 1)² − 1) = 11
- Expand: 4(x − 2)² − 16 + 9(y + 1)² − 9 = 11
- Combine: 4(x − 2)² + 9(y + 1)² = 36
- Divide by 36: (x − 2)²/9 + (y + 1)²/4 = 1
- Center (2, −1), a = 3, b = 2, horizontal major axis.
9. Summary
- Ellipse: set of points where distance-sum to two foci is constant 2a
- Standard form: (x − h)²/a² + (y − k)²/b² = 1, with a > b for horizontal major axis
- c² = a² − b²; foci are on the major axis at distance c from center
- Eccentricity e = c/a measures elongation; circle is e = 0
- Reflective property: rays from one focus pass through the other