The coordinates of the points on the curve `x=a(theta + sintheta), y=a(1-costheta)`, where tangent is inclined an angle `pi/4` to the `x`-axis are- (A) `(a, a)` (B) `(a(pi/2-1),a)` (C) `(a(pi/2+1),a)` (D) `(a,a(pi/2+1))`

Options

A. `(a,a)`
B. `(a(pi//2 -1), a)`
C. `(a(pi//2+1),a)`
D. `(a,a(pi//2 +1))`

Answer

To solve the problem of finding the coordinates of the points on the curve defined by \( x = a(\theta + \sin \theta) \) and \( y = a(1 - \cos \theta) \) where the tangent is inclined at an angle of \( \frac{\pi}{4} \) to the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Slope Condition**:
The slope of the tangent line at the point is given by the angle \( \frac{\pi}{4} \). The tangent of this angle is:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Therefore, the slope \( m \) of the tangent line is:
\[ m = 1 \]

2. **Find the Derivative \( \frac{dy}{dx} \)**:
To find the slope of the tangent line in terms of \( \theta \), we need to compute \( \frac{dy}{dx} \) using the chain rule:
\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \]

3. **Calculate \( \frac{dx}{d\theta} \)**:
Differentiate \( x \) with respect to \( \theta \):
\[ x = a(\theta + \sin \theta) \implies \frac{dx}{d\theta} = a(1 + \cos \theta) \]

4. **Calculate \( \frac{dy}{d\theta} \)**:
Differentiate \( y \) with respect to \( \theta \):
\[ y = a(1 - \cos \theta) \implies \frac{dy}{d\theta} = a \sin \theta \]

5. **Substitute into the Derivative**:
Now substitute \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \) into the derivative:
\[ \frac{dy}{dx} = \frac{a \sin \theta}{a(1 + \cos \theta)} = \frac{\sin \theta}{1 + \cos \theta} \]

6. **Set the Derivative Equal to 1**:
Since we want the slope to be 1:
\[ \frac{\sin \theta}{1 + \cos \theta} = 1 \]

7. **Solve for \( \theta \)**:
Cross-multiplying gives:
\[ \sin \theta = 1 + \cos \theta \]
Rearranging leads to:
\[ \sin \theta - \cos \theta = 1 \]
Dividing both sides by \( \sqrt{2} \):
\[ \frac{\sin \theta}{\sqrt{2}} - \frac{\cos \theta}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]
This can be rewritten using the sine subtraction formula:
\[ \sin\left(\theta - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Thus, we find:
\[ \theta - \frac{\pi}{4} = \frac{\pi}{4} \implies \theta = \frac{\pi}{2} \]

8. **Find the Coordinates**:
Substitute \( \theta = \frac{\pi}{2} \) back into the equations for \( x \) and \( y \):
\[ x = a\left(\frac{\pi}{2} + \sin\left(\frac{\pi}{2}\right)\right) = a\left(\frac{\pi}{2} + 1\right) \]
\[ y = a\left(1 - \cos\left(\frac{\pi}{2}\right)\right) = a(1 - 0) = a \]
Therefore, the coordinates are:
\[ (x, y) = \left(a\left(\frac{\pi}{2} + 1\right), a\right) \]

### Final Answer:
The coordinates of the points on the curve where the tangent is inclined at an angle \( \frac{\pi}{4} \) to the x-axis are:
\[ \left(a\left(\frac{\pi}{2} + 1\right), a\right) \]

This corresponds to option (C): \( (a(\frac{\pi}{2} + 1), a) \).