Lesson 73: Ellipses
An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points called foci is constant. Ellipses appear in planetary orbits, optics, and engineering designs.
Equations of an Ellipse
Standard forms:
- Horizontal major axis: (x − h)² / a² + (y − k)² / b² = 1, center at (h, k), a > b
- Vertical major axis: (x − h)² / b² + (y − k)² / a² = 1, center at (h, k), a > b
Distance from center to foci: c = √(a² − b²)
Examples
Example 1: Find the foci of (x² / 25) + (y² / 16) = 1
Solution: a² = 25, b² = 16 → c = √(25 − 16) = 3 → Foci at (±3, 0)
Example 2: Write the equation of an ellipse with center (0, 0), a = 5, b = 3, vertical major axis
Solution: Equation: x² / 9 + y² / 25 = 1
Example 3: Determine the vertices and foci of (x − 2)² / 9 + (y + 1)² / 4 = 1
Solution: Center = (2, −1), a² = 9 → a = 3, b² = 4 → b = 2 Horizontal major axis → vertices: (2 ± 3, −1) = (−1, −1), (5, −1) Foci: c = √(9 − 4) = √5 → Foci: (2 ± √5, −1)
Practice Problems
- Find the foci of (x² / 49) + (y² / 36) = 1
- Write the equation of an ellipse centered at (1, −2) with a = 6, b = 4, horizontal major axis
- Determine the vertices and foci of (x + 3)² / 16 + (y − 2)² / 9 = 1
- Find c for an ellipse with a = 8, b = 5
- Sketch the ellipse x² / 25 + y² / 9 = 1 showing vertices and foci