Let y = `a(1 - cos theta), x = a(theta- sin theta)`, then y regarded as a function of x is maximum when x equals to

Options

A. `(pi)/(2)`
B. `-(pi)/(2)`
C. `(pi)`
D. `(pi)/(3)`

Answer

To solve the problem, we need to find the maximum value of \( y \) as a function of \( x \) given the parametric equations:

\[ y = a(1 - \cos \theta) \]
\[ x = a(\theta - \sin \theta) \]

### Step 1: Differentiate \( y \) and \( x \) with respect to \( \theta \)

First, we differentiate \( y \) and \( x \) with respect to \( \theta \):

1. **Differentiate \( y \)**:
\[ \frac{dy}{d\theta} = a \cdot \frac{d}{d\theta}(1 - \cos \theta) = a \cdot \sin \theta \]

2. **Differentiate \( x \)**:
\[ \frac{dx}{d\theta} = a \cdot \frac{d}{d\theta}(\theta - \sin \theta) = a \cdot (1 - \cos \theta) \]

### Step 2: Find \( \frac{dy}{dx} \)

Using the chain rule, we can express \( \frac{dy}{dx} \) as follows:

\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{a \sin \theta}{a(1 - \cos \theta)} = \frac{\sin \theta}{1 - \cos \theta} \]

### Step 3: Set \( \frac{dy}{dx} = 0 \)

To find the maximum, we set \( \frac{dy}{dx} = 0 \):

\[ \frac{\sin \theta}{1 - \cos \theta} = 0 \]

This implies that:

\[ \sin \theta = 0 \]

### Step 4: Solve for \( \theta \)

The solutions for \( \sin \theta = 0 \) are:

\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]

### Step 5: Find the corresponding \( x \) values

We need to find the value of \( x \) when \( \theta = \pi \):

\[ x = a(\pi - \sin \pi) = a(\pi - 0) = a\pi \]

### Step 6: Check the nature of the critical point

To confirm that this is a maximum, we can check the sign of \( \frac{dy}{dx} \) around \( \theta = \pi \):

- For \( \theta < \pi \), \( \sin \theta > 0 \) and \( 1 - \cos \theta > 0 \) ⇒ \( \frac{dy}{dx} > 0 \)
- For \( \theta > \pi \), \( \sin \theta < 0 \) and \( 1 - \cos \theta > 0 \) ⇒ \( \frac{dy}{dx} < 0 \)

Since \( \frac{dy}{dx} \) changes from positive to negative at \( \theta = \pi \), this confirms that \( x = a\pi \) is indeed a maximum.

### Final Answer

Thus, the value of \( x \) at which \( y \) is maximum is:

\[ \boxed{a\pi} \]