Question

Distance between the points $(a\cos (\theta +\frac{2\pi }{3}),a\sin (\theta +\frac{2\pi }{3}))$ and $(a\cos (\theta +\frac{\pi }{3}),a\sin (\theta +\frac{\pi }{3}))$

• A. $2a$
• B. $3a$
• C. $\frac{a}{2}$
• D. $a$

Answer 1

The correct option is D $a$

Let $A\left(acos\right(\theta +\frac{2\pi }{3}),asin(\theta +\frac{2\pi }{3}\left)\right)$

$B\left(acos\right(\theta +\frac{\pi }{3}),asin(\theta +\frac{\pi }{3}\left)\right)$

$AB=\sqrt{a^2\left[cos\right(\theta +\frac{\pi }{3})-cos(\theta +\frac{2\pi }{3}){]}^2+a^2[sin(\theta +\frac{\pi }{3})-sin(\theta +\frac{2\pi }{3}){]}^2}$

$=a\sqrt{co{s}^2(\theta +\frac{\pi }{3})+si{n}^2(\theta +\frac{\pi }{3})+co{s}^2(\theta +\frac{2\pi }{3})+si{n}^2(\theta +\frac{2\pi }{3})}$

$-2\left[cos\right(\theta +\frac{\pi }{3}\left)cos\right(\theta +\frac{2\pi }{3}\left)sin\right(\theta +\frac{2\pi }{3}\left)sin\right(\theta +\frac{2\pi }{3}\left)\right]$

$=a\sqrt{1+1-2cos(\theta +\frac{2\pi }{3}-\theta -\frac{\pi }{3})}$

$=a\sqrt{2-2cos\frac{\pi }{3}}$

$=a\sqrt{2-2\frac{1}{2}}=a\sqrt{2-1}=a\sqrt{1}=a$