If ` x=a cos theta ,y=asin theta ,then (dy)/(dx)=`

Options

A. ` x^(3) y`
B. ` -tan theta `
C. `cot theta `
D. `-cot theta `

Answer

To find \(\frac{dy}{dx}\) given the equations \(x = a \cos \theta\) and \(y = a \sin \theta\), we will use the chain rule of differentiation. Here’s the step-by-step solution:

### Step 1: Differentiate \(x\) with respect to \(\theta\)
Given:
\[ x = a \cos \theta \]
We differentiate \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = -a \sin \theta \]

### Step 2: Differentiate \(y\) with respect to \(\theta\)
Given:
\[ y = a \sin \theta \]
We differentiate \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = a \cos \theta \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]
Substituting the derivatives we found in Steps 1 and 2:
\[ \frac{dy}{dx} = \frac{a \cos \theta}{-a \sin \theta} \]

### Step 4: Simplify the expression
The \(a\) in the numerator and denominator cancels out:
\[ \frac{dy}{dx} = \frac{\cos \theta}{-\sin \theta} = -\cot \theta \]

### Final Answer
Thus, we have:
\[ \frac{dy}{dx} = -\cot \theta \]

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