The angle between the radius vector and tangent at any point of the curve $r=a(1-cos~ heta)$ is :

Question

The angle between the radius vector and tangent at any point of the curve $r=a(1-cos~	heta)$ is :

Accepted Answer

To determine the angle between the radius vector and the tangent at any point of the curve given by the polar equation $ r = a(1 - \cos \theta) $, we need to analyze the geometry of the curve and the relationship between the radius vector and the tangent line.

### Explanation for Option 3: $\theta/2$

1. **Curve Analysis**: The given curve $ r = a(1 - \cos \theta) $ is a well-known polar equation for a cardioid. The cardioid is a type of limaçon and has a distinct shape where the curve is symmetric about the polar axis.

2. **Radius Vector and Tangent**: At any point on the curve, the radius vector is the line segment from the origin to the point $(r, \theta)$. The tangent to the curve at this point is the line that just touches the curve at this point and is perpendicular to the radius vector.

3. **Angle Calculation**: The angle between the radius vector and the tangent line can be found using the derivative of the curve in polar coordinates. The slope of the tangent line in polar coordinates is given by:
   $$
   \frac{dr}{d\theta} = a \sin \theta
   $$
   The angle $\phi$ between the radius vector and the tangent line is given by:
   $$
   \tan \phi = \frac{r \frac{d\theta}{dr}}{r} = \frac{a \sin \theta}{a(1 - \cos \theta)}
   $$
   Simplifying, we find:
   $$
   \tan \phi = \frac{\sin \theta}{1 - \cos \theta} = \cot \frac{\theta}{2}
   $$
   Therefore, $\phi = \frac{\theta}{2}$.

4. **Conclusion**: The angle between the radius vector and the tangent at any point on the cardioid is $\frac{\theta}{2}$, which confirms that Option 3 is correct.

### Explanation for Incorrectness of Other Options

- **Option 1: $\theta$**: This option suggests that the angle between the radius vector and the tangent is equal to the polar angle $\theta$ itself. However, as derived, the angle is actually $\frac{\theta}{2}$, not $\theta$.

- **Option 2: $2\theta$**: This option implies that the angle is twice the polar angle $\theta$. This is incorrect because the derivation shows that the angle is half of $\theta$, not double.

- **Option 4: $\frac{3\theta}{2}$**: This option suggests that the angle is $\frac{3}{2}$ times the polar angle $\theta$. This is incorrect as the correct angle is $\frac{\theta}{2}$, not $\frac{3\theta}{2}$.

In conclusion, the correct angle between the radius vector and the tangent at any point on the curve $ r = a(1 - \cos \theta) $ is $\frac{\theta}{2}$, making Option 3 the correct choice.