The normal to the curve `x=a(cos theta + theta sin theta), y=a(sin theta - theta cos theta)` at any `theta` is such that

Options

A. it makes a constant angle with x-axis
B. it passes through the origin
C. it is at a constant distance from the origin
D. none of these

Answer

To solve the problem, we need to find the normal to the curve given by the parametric equations:

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

at any point defined by the parameter \(\theta\). We will derive the equation of the normal line and analyze its properties.

### Step 1: Find \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\)

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

\[ \frac{dx}{d\theta} = -a \sin \theta + a \sin \theta + a \theta \cos \theta = a \theta \cos \theta \]

\[ \frac{dy}{d\theta} = a \cos \theta - a \cos \theta + a \theta \sin \theta = a \theta \sin \theta \]

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

Now, we can find \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a \theta \sin \theta}{a \theta \cos \theta} = \tan \theta \]

### Step 3: Find the slope of the normal

The slope of the tangent line is \(\tan \theta\), so the slope of the normal line, \(m\), is given by:

\[ m = -\cot \theta \]

### Step 4: Write the equation of the normal line

Using the point-slope form of the line, the equation of the normal line at the point \((x_1, y_1)\) is:

\[ y - y_1 = m(x - x_1) \]

Substituting \(x_1\) and \(y_1\) with the expressions for \(x\) and \(y\):

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

### Step 5: Rearranging the equation

Rearranging the equation gives us:

\[ y - a \sin \theta + a \theta \cos \theta = -\frac{\cos \theta}{\sin \theta} \left(x - a \cos \theta - a \theta \sin \theta\right) \]

Multiplying through by \(\sin \theta\) to eliminate the fraction:

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

### Step 6: Simplifying the equation

After simplifying, we arrive at:

\[ x \cos \theta + y \sin \theta = a \cos^2 \theta + a \sin^2 \theta \]

Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\):

\[ x \cos \theta + y \sin \theta = a \]

### Conclusion

The normal line to the curve at any point defined by \(\theta\) has the equation:

\[ x \cos \theta + y \sin \theta = a \]

### Analyzing the Properties

1. **Constant Angle with x-axis**: The angle \(\theta\) varies, so this is not constant.
2. **Passes through the Origin**: This is not true for all \(\theta\).
3. **Constant Distance from the Origin**: The distance from the origin is constant as it equals \(a\).
4. **None of these**: This is incorrect since we have a valid condition.

Thus, the correct answer is that the normal is at a constant distance from the origin.