点P是椭圆x^2/a^2+y^2/b^2=1任一点，O为坐标原点。（1）OP与X轴正半轴成α角求，模OP.

P(acosx,bsinx)

tanα=bsinx/acosx = b/a * tanx

tanx = a/b * tanα

tan^2 x = a^2/b^2 * tan^2 α

cos^2 x = 1/(a^2/b^2 * tan^2 α + 1)

= b^2 / (a^2tan^2 α + b^2)

sin^2 x = a^2tan^2 α / (a^2tan^2 α + b^2)

模OP = 根号(a^2(b^2 / (a^2tan^2 α + b^2)) + b^2(a^2tan^2 α / (a^2tan^2 α + b^2)))

= 根号(a^2b^2 / (a^2tan^2 α + b^2) + a^2b^2tan^2 α / (a^2tan^2 α + b^2)))

= 根号(a^2b^2(1+tan^2 α)/ (a^2tan^2 α + b^2))

= ab根号(1/(a^2tan^2α + b^2))/cosα

α'=α+pi/2

1/(模OP^2) +1/(模QO^2)

= cos^2α(a^2tan^2α + b^2)/a^2b^2 + cos^2α'(a^2tan^2α' + b^2)/a^2b^2

= (cos^2α(a^2tan^2α + b^2) + sin^2α(a^2cot^2α + b^2))/a^2b^2

= (a^2tan^2αcos^2α + b^2cos^2α + a^2sin^2αcot^2α + b^2sin^2α))/a^2b^2

= (a^2sin^2α + b^2cos^2α + a^2cos^2α + b^2sin^2α))/a^2b^2

= (a^2*1 + b^2*1)/a^2b^2

= (a^2 + b^2)/a^2b^2

所以1/(模OP^2) +1/(模Q