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

$$\mathrm{If}{\mathrm{x}}^2+{\mathrm{y}}^2=1\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{maximum}$$$$\mathrm{value}\mathrm{of}{\mathrm{x}}^2+4\mathrm{xy}-{\mathrm{y}}^2.$$

Answered by iloveisrael last updated on 27/Aug/21

$$let\{\begin{array}{l}x=\cos \theta \\ y=\sin \theta \end{array}$$$$f\left(\theta \right)=\cos {}^2\theta +4\cos \theta \sin \theta -\sin {}^2\theta$$$$f\left(\theta \right)=\frac{1+\cos 2\theta }{2}+2\sin 2\theta -\left(\frac{1-\cos 2\theta }{2}\right)$$$$f\left(\theta \right)=\cos 2\theta +2\sin 2\theta$$$$f\left(\theta \right)=\sqrt{5}\cos (2\theta -{\tan }^{-1}\left(2\right))$$$$maxf\left(\theta \right)=\sqrt{5},when2\theta ={\tan }^{-1}\left(2\right)+2n\pi$$$$\theta =\frac{1}{2}{\tan }^{-1}\left(2\right)+n\pi ,n\in \mathbb{Z}$$

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