Question

椭圆C:x^2/4+y^2/3=1,A1,A2是长轴的两端点,

Answer 1

设P(2cosa,√3sina),A1(-2,0),A2(2,0) 右准线 x=4

直线 PA1 ： y-0 = (x+2) *[√3sina/(2cosa+2)] = (x+2)*[2√3sin(a/2)cos(a/2)]/[4(cosa/2)^2]

=√3(x+2) tan(a/2)/2 令k = tan(a/2)

y=√3(x+2)k/2

因此 yM = 3√3k; M(4,3√3k) MF1: (5,3√3k)

直线 PA2 y-0 = (x-2)*[√3sina/(2cosa-2)]=(x-2)*[2√3sin(a/2)cos(a/2)]/[-4(sina/2)^2]

= -√3(x-2)cot(a/2) /2

y = -√3(x-2)/2k

yN = -√3/k ; N(4,-√3/k) NF2 (3,-√3/k)

因此 MF1*NF2 = 15 +27 =42