y= (( )/(3)+ ^-1(x)/(2))(x-y)^2+3y^2 Answer: 3

Question

$y=\cos (\frac{\pi }{3}+{\cos }^{-1}\frac{x}{2})(x-y{)}^{2}+3{y}^{2}$

Answer: 3

Accepted Answer

$\begin{gathered} y=\cos (\frac{\pi }{3}+{\cos }^{-1}\frac{x}{2}) \ =\cos (\frac{\pi }{3})\cos ({\cos }^{-1}(\frac{x}{2}))-\sin (\frac{\pi }{3.})\sin ({\cos }^{-1}(\frac{x}{2})) \ =\frac{1}{2}\cdot \frac{x}{2}-\frac{\sqrt{3}}{2}\cdot \sqrt{1-\frac{{x}^{2}}{4}} \ \Rightarrow 4y=x-\sqrt{3}\sqrt{4-{x}^{2}} \ \Rightarrow (4y-x{)}^{2}=3(4-{x}^{2}) \ \Rightarrow 16{y}^{2}+{x}^{2}-8xy=12-3{x}^{2} \ {x}^{2}+4{y}^{2}-2xy=3 \ (x-y{)}^{2}+3{y}^{2}=3 \end{gathered}$

Solutions · 1