Question

已知椭圆X^2/a^2+y^2/b^2=1上任意一点M与短轴两端点B1,B2的连线分别与X轴交于P,Q两点,O为椭圆的中

Answer 1

设任一点M(acost,bsint)

短轴两端点A(0,b),B(0,-b)

MA交x轴于P(x1,0),MB交x轴于Q(x2,0)

b/x1=(b-bsint)/acost

x1=acost/(1-sint)

bsint/(acost-x2)=b/x2

x2=acost/(1+sint)

|OP|*|OQ|=|x1|*|x2|=a^2cos^2t/(1-sint)(1+sint)

=a^2

所以|OP|*|OQ|为定值.