Question

If the normal at any point P on ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2)) =1` meets the auxiliary circle at Q and R such that `/_QOR = 90^(@)` where O is centre of ellipse, then
A. `a^(4) +2b^(3) ge 3a^(2)b^(2)`
B. `a^(4) +2b^(4) ge 5a^(2)b^(2)+2a^(3)b`
C. `a^(4)+2b^(4) ge 3a^(2)b^(2)+ab`
D. None of these

Answer 1

Correct Answer - B
Normal at `P(a cos theta, b sin theta)` is
`ax sec theta - by cosec theta = a^(2) -b^(2)`
Homogenising with auxilliary circle
`x^(2) + y^(2) = a^(2)`
`x^(2) + y^(2) = (a)^(2) ((ax sec theta - by cosec theta)^(2))/((a^(2)-b^(2))^(2))`
`:.` For `/_QOR = 90^(@)`
Coefficient of `x^(2)+` Coefficient of `y^(2) =0`
`1- (a^(4)sec^(4)theta)/((a^(2)-b^(2))^(2)) + 1-(a^(2)b^(2)cosec^(2)theta)/((a^(2)-b^(2))^(2)) =0`
`a^(4) - 5a^(2)b^(2) + 2b^(4) = a^(4) tan^(2) theta + a^(2)b^(2) cot^(2) theta`
`:. AM ge GM`
`a^(4) - 5a^(2)b^(2)+2b^(4) ge 2a^(3)b`