If \(x = a\sin 2t(1 + \cos 2t)\) and \(y = b\cos 2t(1 - \cos 2t)\), then \(\left(\displaystyle\frac{dy}{dx}\right)_{\text{at}\,t=\frac{\pi}{4}}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{b}{a}\)
For a curve in parametric form, \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\). Expanding with \(2\sin\theta\cos\theta = \sin 2\theta\) and \(2\cos^2\theta = 1 + \cos 2\theta\):
\( \displaystyle x = a\sin 2t + \frac{a}{2}\sin 4t \)
\( \displaystyle y = b\cos 2t - \frac{b}{2}(1 + \cos 4t) \)
Differentiating with respect to t:
\( \displaystyle \frac{dx}{dt} = 2a\cos 2t + 2a\cos 4t \)
\( \displaystyle \frac{dy}{dt} = -2b\sin 2t + 2b\sin 4t \)
At \(t = \frac{\pi}{4}\), \(2t = \frac{\pi}{2}\) and \(4t = \pi\):
\( \displaystyle \frac{dx}{dt} = 2a(0) + 2a(-1) = -2a \)
\( \displaystyle \frac{dy}{dt} = -2b(1) + 2b(0) = -2b \)
\( \displaystyle \frac{dy}{dx} = \frac{-2b}{-2a} = \frac{b}{a} \)
NCERT Exemplar:
Continuity and Differentiability
