If-x-2-y-2-1-then-find-the-maximum-value-of-x-2-4xy-y-2-

Question Number 152340 by imjagoll last updated on 27/Aug/21

If x^{2} + y^{2} = 1 then find the maximum

value of x^{2} + 4 xy - y^{2} .

Answered by iloveisrael last updated on 27/Aug/21

l e t {\begin{cases} x = \cos θ \\ y = \sin θ \end{cases}

f (θ) = \cos^{2} θ + 4 \cos θ \sin θ - \sin^{2} θ

f (θ) = \frac{1 + \cos 2 θ}{2} + 2 \sin 2 θ - (\frac{1 - \cos 2 θ}{2})

f (θ) = \cos 2 θ + 2 \sin 2 θ

f (θ) = \sqrt{5} \cos (2 θ - \tan^{- 1} (2))

m a x f (θ) = \sqrt{5}, w h e n 2 θ = \tan^{- 1} (2) + 2 n π

θ = \frac{1}{2} \tan^{- 1} (2) + n π, n \in Z