A ray of light makes an angle of `20^(@)` with the horizontal and strikes a plane mirror which is inclined at an angle ` theta` to the horizontal. The angle ` theta`for which the reflected ray becomes vertical, is

Options

A. `40^(@)`
B. `80^(@)`
C. `55^(@) & 35^(@)`
D. `100^(@)`

Answer

To solve the problem, we need to determine the angle \( \theta \) for which the reflected ray becomes vertical when a ray of light strikes a plane mirror inclined at angle \( \theta \) to the horizontal. The incident ray makes an angle of \( 20^\circ \) with the horizontal.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- The incident ray makes an angle of \( 20^\circ \) with the horizontal.
- The plane mirror is inclined at an angle \( \theta \) to the horizontal.
- The reflected ray is vertical, meaning it makes an angle of \( 90^\circ \) with the horizontal.

2. **Identifying the Angles**:
- The angle of incidence \( i \) is the angle between the incident ray and the normal to the mirror.
- The angle of reflection \( r \) is the angle between the reflected ray and the normal to the mirror.
- According to the law of reflection, \( i = r \).

3. **Finding the Angle of Incidence**:
- The angle between the horizontal and the normal to the mirror is \( \theta \).
- Therefore, the angle of incidence \( i \) can be expressed as:
\[ i = 90^\circ - \theta - 20^\circ = 70^\circ - \theta \]

4. **Finding the Angle of Reflection**:
- Since the reflected ray is vertical, the angle of reflection \( r \) is:
\[ r = 90^\circ \]

5. **Setting Up the Equation**:
- According to the law of reflection, we have:
\[ i = r \]
- Substituting the values we found:
\[ 70^\circ - \theta = 90^\circ \]

6. **Solving for \( \theta \)**:
- Rearranging the equation gives:
\[ \theta = 70^\circ - 90^\circ = -20^\circ \]
- This negative angle indicates that we need to consider the other case where the ray is inclined downwards.

7. **Considering the Second Case**:
- If the incident ray is inclined downwards at \( 20^\circ \), the angle of incidence \( i \) becomes:
\[ i = 90^\circ + 20^\circ - \theta = 110^\circ - \theta \]
- The angle of reflection remains the same:
\[ r = 90^\circ \]
- Setting up the equation again:
\[ 110^\circ - \theta = 90^\circ \]

8. **Solving for \( \theta \) Again**:
- Rearranging gives:
\[ \theta = 110^\circ - 90^\circ = 20^\circ \]

9. **Final Calculation**:
- The total angle from the horizontal to the mirror is:
\[ \theta + 20^\circ = 90^\circ \]
- Therefore, the angle \( \theta \) for which the reflected ray becomes vertical is:
\[ \theta = 35^\circ \text{ or } 55^\circ \]

### Conclusion:
The angles \( \theta \) for which the reflected ray becomes vertical are \( 35^\circ \) and \( 55^\circ \).

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