Lesson 74: Hyperbolas
A hyperbola is the set of all points in a plane such that the absolute difference of the distances from two fixed points called foci is constant. Hyperbolas appear in satellite navigation, radio antennas, and certain types of orbits.
Equations of a Hyperbola
Standard forms:
- Horizontal transverse axis: (x − h)² / a² − (y − k)² / b² = 1, center at (h, k)
- Vertical transverse axis: (y − k)² / a² − (x − h)² / b² = 1, center at (h, k)
Distance from center to foci: c = √(a² + b²)
Asymptotes (for hyperbola centered at (h, k)):
- Horizontal: y − k = ±(b/a)(x − h)
- Vertical: y − k = ±(a/b)(x − h)
Examples
Example 1: Find the foci of (x² / 16) − (y² / 9) = 1
Solution: a² = 16 → a = 4, b² = 9 → b = 3 c = √(16 + 9) = √25 = 5 → Foci: (±5, 0)
Example 2: Write the equation of a vertical hyperbola centered at (0, 0) with a = 3 and b = 4
Solution: Equation: y² / 9 − x² / 16 = 1
Example 3: Find the equations of the asymptotes for (x − 1)² / 25 − (y + 2)² / 9 = 1
Solution: Horizontal transverse axis → Asymptotes: y + 2 = ±(3/5)(x − 1) → y = −2 ± 0.6(x − 1)
Practice Problems
- Find the foci of (x² / 36) − (y² / 16) = 1
- Write the equation of a hyperbola with center at (2, −1), vertical transverse axis, a = 5, b = 3
- Find the asymptotes of (y − 3)² / 9 − (x + 2)² / 16 = 1
- Determine c for a hyperbola with a = 7, b = 24
- Sketch the hyperbola x² / 9 − y² / 4 = 1 showing vertices, foci, and asymptotes