If \(x = 3\cos t\) and \(y = 2\sin t\), then the value of \(\displaystyle\frac{dy}{dx}\) at \(t = \displaystyle\frac{\pi}{4}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(-\displaystyle\frac{2}{3}\)
In parametric form, \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\):
\( \displaystyle \frac{dx}{dt} = -3\sin t, \quad \frac{dy}{dt} = 2\cos t \)
\( \displaystyle \frac{dy}{dx} = \frac{2\cos t}{-3\sin t} = -\frac{2}{3}\cot t \)
At \(t = \frac{\pi}{4}\), \(\cot t = 1\):
\( \displaystyle \frac{dy}{dx} = -\frac{2}{3} \)