Find the maximum distance of any point on the curve `x^2+2y^2+2x y=1` from the origin.

Explanation

Let the polar coordinates of any point on the curve be `P(r,theta)`.
Then its Cartesian coordinates are `P(r cos theta,rsin theta)`.
Point P lies on the curve. Then
`r^2cos^2theta+2r^2sin^2thetacostheta=1`
or `r^2=(1)/(cos^2theta+2sin^2theta+sin2theta)`
`=(1)/(sin^2theta+1+sin2theta)`
`=(2)/(1-cos2theta+2+2sin2theta)`
`=(2)/(3-cos2theta +2sin2theta) `
Now, `-sqrt(5)le-cos2theta +2sin2theta le sqrt(5)`
or `3-sqrt(5) le -cos 2theta+2sin2theta le 3+sqrt(5)`
or `r_("max")^2 =(2)/(3-sqrt(5)`
or `r_("max")=(sqrt(2))/(sqrt(3-sqrt5))`