求椭圆5x2+8xy+5y2=9的长半轴与短半轴

Question

求椭圆5x2+8xy+5y2=9的长半轴与短半轴

Answer 1

在原式中用-x和-y取代x和y, 原式不变,说明原点为椭圆的对称中心,长短轴均过原点.
取直线y = kx, 其与椭圆的交点为A, B.
该直线与椭圆长轴重合时,AB间的距离(d)最大； 与椭圆短轴重合时,AB间的距离(d)最小.
将y = kx代入椭圆方程： 5x² + 8x*kx + 5(kx)² = 9
(5k² + 8x + 5)x² = 9
x = 3/√(5k² + 8x + 5), y = 3k/√(5k² + 8x + 5), A(3/√(5k² + 8x + 5), 3k/√(5k² + 8x + 5))
x = -3/√(5k² + 8x + 5), y = -3k/√(5k² + 8x + 5), B(-3/√(5k² + 8x + 5), -3k/√(5k² + 8x + 5))
f(k) = AB² = [6/√(5k² + 8x + 5)]² + [6k/√(5k² + 8x + 5)]² = 36(k² + 1)/(5k² + 8x + 5)
f'(k) = 36*2k/(5k² + 8x + 5) + 36(k² +1)(-1)(10k + 8)/(5k² + 8x + 5)²
= 36*8(k² -1)/(5k² + 8x + 5)² = 0
k = 1或k = -1
k = 1时,AB² = 4,AB = 2, 短半轴 = 1
k = -1时,AB² = 36,AB = 6, 长半轴 = 3