Hyperbola: The locus of points for which the absolute value of the difference between the distances from two foci F and F′ is constant (2a).
Standard form x²/a² − y²/b² = 1 (a, b > 0):
・Vertices (±a, 0) ・Foci (±c, 0), c² = a² + b²
・Asymptotes y = ±(b/a)x
Example ① Find the foci and asymptotes of x²/9 − y²/16 = 1.
Solution:a=3, b=4, c=5。Foci (±5, 0), asymptotes y = ±(4/3)x
Solution:a=3, b=4, c=5。Foci (±5, 0), asymptotes y = ±(4/3)x
Key Point
- Hyperbola: constant difference; ellipse: constant sum
Practice Problems
- Find the foci of x²/16 − y²/9 = 1.
Answers and Explanations
- Answer:(±5, 0)