When two plane mirrors subtend an angle `theta` between them, then a ray of light incident parallel to one of them retraces its path afer n reflections such that `ntheta=c` (where c is constant)
The graph between `theta` and n will

Options

A. be continous but not differentiable
B. be continuous & differentiable
C. Not be continous
D. Can't say

Answer

To solve the problem, we need to analyze the relationship between the angle \( \theta \) between two mirrors and the number of reflections \( n \) that a ray of light undergoes before retracing its path. The key relationship given is \( n\theta = c \), where \( c \) is a constant.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have two plane mirrors that subtend an angle \( \theta \) between them.
- A ray of light is incident parallel to one of the mirrors.

2. **Analyzing Reflections**:
- For \( n = 1 \):
- The ray reflects off the first mirror and retraces its path, which requires \( \theta = 90^\circ \) (since the ray must be parallel to the other mirror).
- For \( n = 2 \):
- The ray reflects twice. However, there is no angle \( \theta \) that allows the ray to retrace its path after two reflections. Thus, \( \theta \) does not exist for \( n = 2 \).
- For \( n = 3 \):
- The ray reflects three times. The first reflection occurs at angle \( \theta \), the second reflection at angle \( 90^\circ - \theta \), and for the ray to retrace its path, the third reflection must also be at \( 90^\circ \).
- This gives us the equation: \( 90 - \theta + \theta + \theta = 180 \), leading to \( 2\theta = 90 \) or \( \theta = 45^\circ \).

3. **Finding a Pattern**:
- Continuing this process, we can find values of \( \theta \) for other odd values of \( n \) (like \( n = 5, 7, \) etc.) and notice that \( \theta \) decreases as \( n \) increases.
- For even values of \( n \), \( \theta \) does not exist.

4. **Graphing the Relationship**:
- We can plot the values of \( \theta \) against \( n \):
- For \( n = 1 \), \( \theta = 90^\circ \).
- For \( n = 2 \), no value exists.
- For \( n = 3 \), \( \theta = 45^\circ \).
- For \( n = 4 \), no value exists.
- For \( n = 5 \), we can find another value, and so forth.
- The graph will show points for odd \( n \) and gaps for even \( n \).

5. **Conclusion**:
- The graph between \( \theta \) and \( n \) will be discontinuous since there are values of \( n \) (even integers) for which \( \theta \) does not exist.

### Final Graph:
- The graph will consist of discrete points corresponding to odd values of \( n \) and will be discontinuous at even values of \( n \).