A ray of light makes an angle of `10^@` 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. `50^(@)`
C. `80^(@)`
D. `100^(@)`

Explanation

To solve the problem, we need to find the angle \( \theta \) at which the plane mirror must be inclined so that the reflected ray becomes vertical. Let's break down the solution step by step.

### Step 1: Understand the given angles
The ray of light makes an angle of \( 10^\circ \) with the horizontal. This means that the angle of incidence \( i \) with respect to the normal to the mirror is:
\[ i = 90^\circ - 10^\circ = 80^\circ \]

**Hint:** Remember that the angle of incidence is measured from the normal, not the horizontal.

### Step 2: Relate the angle of incidence and reflection
According to the law of reflection, the angle of incidence is equal to the angle of reflection. Therefore, the angle of reflection \( r \) is:
\[ r = i = 80^\circ \]

**Hint:** The law of reflection states that the angle of incidence equals the angle of reflection.

### Step 3: Determine the angle of the reflected ray with respect to the mirror
The angle between the reflected ray and the plane of the mirror is given by:
\[ \text{Angle with mirror} = 90^\circ - r = 90^\circ - 80^\circ = 10^\circ \]

**Hint:** The angle with the mirror is complementary to the angle of reflection.

### Step 4: Find the angle of the reflected ray with respect to the horizontal
Since the mirror is inclined at an angle \( \theta \) to the horizontal, the angle of the reflected ray with respect to the horizontal will be:
\[ \text{Angle with horizontal} = \theta + 10^\circ \]

**Hint:** When adding angles, ensure you account for the inclination of the mirror.

### Step 5: Set the angle of the reflected ray to be vertical
For the reflected ray to be vertical, its angle with the horizontal must be \( 90^\circ \). Therefore, we set up the equation:
\[ \theta + 10^\circ = 90^\circ \]

**Hint:** A vertical ray has an angle of \( 90^\circ \) with the horizontal.

### Step 6: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = 90^\circ - 10^\circ = 80^\circ \]

### Final Answer
Thus, the angle \( \theta \) for which the reflected ray becomes vertical is:
\[ \theta = 80^\circ \]

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