如图,经过点P(2,3),且中点在坐标原点,焦点在x轴上的椭圆上,椭圆M的离心率为0.5

e² = c²/a² = (a² - b²)/a² = 1 - b²/a² = 1/4,b² = 3a²/4
x²/a² + 4y²/(3a²) = 1
过P(2,3):4/a² + 36/(3a²) = 1
a² = 16,b² = 12
x²/16 + y²/12 = 1
设PA斜率为k,显然PB的斜率为-k
PA:y - 3 = k(x - 2),y = kx + 3 - 2k
与x²/16 + y²/12 = 1联立,可得一元二次方程,其一解为x = 2,另一解为(A):
x = (8k² - 24k -6)/(4k² + 3),y = 3 - (24k² + 12k)/(4k² + 3)
PB:y - 3 = -k(x - 2)
与x²/16 + y²/12 = 1联立,可得一元二次方程,其一解为x = 2,另一解为(B):
x = (8k² + 24k -6)/(4k² + 3),y = 3 - (24k² - 12k)/(4k² + 3)
两点横坐标之差 ∆x = 48k/(4k² + 3)
两点纵坐标之差 ∆y = 24k/(4k² + 3)
AB斜率为:∆y/(∆x) = 1/2