If y=(3+^-1x2), then (x-y)^2+3y^2 is equal to _______.

Question

If $y=\cos\left(\dfrac{\pi}{3}+\cos^{-1}\dfrac{x}{2}\right)$, then $(x-y)^2+3y^2$ is equal to \_\_\_\_\_\_\_.

Answer: 3

Accepted Answer

$$y=\cos\frac{\pi}{3}\cos\left(\cos^{-1}\frac{x}{2}\right)-\sin\frac{\pi}{3}\sin\left(\cos^{-1}\frac{x}{2}\right)=\frac12\cdot\frac{x}{2}-\frac{\sqrt3}{2}\sqrt{1-\frac{x^2}{4}}.$$

$$\Rightarrow 4y=x-\sqrt3\sqrt{4-x^2}\ \Rightarrow\ (4y-x)^2=3(4-x^2)\ \Rightarrow\ 16y^2+x^2-8xy=12-3x^2.$$

$$\Rightarrow x^2+4y^2-2xy=3\ \Rightarrow\ (x-y)^2+3y^2=3.$$

Solutions · 1