Question

如图,求椭圆C的离心率,

Answer 1

x²/a² + 3x²/b² = 1

(b² + 3a²)x² = a²b²

x = ±ab/√(3a² + b²)

不妨记t = ab/√(3a² + b²)

A(t,-√3t),B(-t,√3t)

F(c,0)

FA的斜率p = (0 + √3t)/(c-t) = √3t/(c-t)

FB的斜率q = (0 - √3t)/(c + t) = -√3t/(c+t)

pq = -1 = -3t²/(c² - t²)

3t² = c² - t²

4t² = c²

4a²b²/(3a² + b²) = c²

4b²/(3a² + b²) = c²/a² = e²

e² = 4(a² - c²)/(3a² + a² - c²) = 4(a² - c²)/(4a² - c²) = 4(1 - e²)/(4 - e²)

4e² - e⁴ = 4 - 4e²

e⁴ - 8e² + 4 = 0

e² = (8 - 4√3)/2 = 4 - 2√3 = (√3 - 1)²

e = √3 - 1