Question

If \(x = a \cos \theta\), \(y = a \sin \theta\), find \(\frac{dy}{dx}\)

• A. \(-\tan \theta\)
• B. \(-\cot \theta\)
• C. \(\tan \theta\)
• D. \(\cot \theta\)

Hint:

The given equations define a circle \(x^2 + y^2 = a^2\). Differentiating implicitly: \(2x + 2y\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y} = -\frac{a \cos\theta}{a \sin\theta} = -\cot\theta\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Parametric differentiation is applied when both \(x\) and \(y\) are defined in terms of a third independent variable or parameter \(\theta\).
Key Formula or Approach:
\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \]
provided \(\frac{dx}{d\theta} \neq 0\).

Step 2: Detailed Explanation:
Differentiating \(x\) with respect to \(\theta\):
\[ x = a \cos \theta \implies \frac{dx}{d\theta} = -a \sin \theta \]
Differentiating \(y\) with respect to \(\theta\):
\[ y = a \sin \theta \implies \frac{dy}{d\theta} = a \cos \theta \]
Applying the chain rule for parametric equations:
\[ \frac{dy}{dx} = \frac{a \cos \theta}{-a \sin \theta} = -\frac{\cos \theta}{\sin \theta} = -\cot \theta \]

Step 3: Final Answer:
Therefore, the derivative \(\frac{dy}{dx}\) is \(-\cot \theta\), matching option (B).