Question

The angle of intersection of the curves $y=2si{n}^{2}xandy=cos2xatx=\frac{\pi }{6}$ is

• A. π/4
• B. π/3
• C. π/2
• D. None of these

Answer 1

Answer: (B), (D)

The correct options are
B

π/3

D

None of these

We have,
$y=si{n}^{2}x...\left(1\right)y=cos2x...\left(2\right)$ And
On differentiating equation (1) w.r.t x, we get
$\frac{dy}{dx}=4sinxcosx{\left[\frac{dy}{dx}\right]}_{x-\frac{\pi }{6}}=4\left(\frac{1}{2}\right)\frac{\sqrt{3}}{2}=\sqrt{3}=m_1\left(say\right)$
On differentiating equation (2) w.r.t x, we get
$\frac{dy}{dx}=-2sin2x{\left[\frac{dy}{dx}\right]}_{x-\frac{\pi }{6}}=-2sin\frac{\pi }{3}=-\sqrt{3}=m_2\left(say\right)$
Hence, angle between the two curves is
$\theta =\pm ta{n}^{-1}\left(\frac{m_1-m_2}{1+m_1{m}_2}\right)=\pm ta{n}^{-1}\sqrt{3}=\frac{\pi }{3}or\frac{2\pi }{3}$
Hence (b) is the correct answer.