A ray is travelling along x-axis in negative x-direction. A plane mirror is placed at origin facing the ray. What should be the angle of plane mirror with the x-axis so that the ray of light after reflecting from the plane mirror passes through point `(1m, sqrt(3)m)` ?

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. `90^(@)`

Answer

To solve the problem step by step, we need to determine the angle at which the plane mirror should be placed so that a ray traveling in the negative x-direction reflects off the mirror and passes through the point (1m, √3m).

### Step 1: Understand the Setup
- A ray is traveling along the negative x-axis towards the origin where the plane mirror is placed.
- The mirror is positioned at an angle θ with respect to the x-axis.
- After reflecting off the mirror, the ray should pass through the point (1m, √3m).

### Step 2: Determine the Direction of the Ray After Reflection
- The coordinates of the point (1m, √3m) indicate that the ray must have a certain angle of elevation after reflection.
- The y-coordinate (√3m) and the x-coordinate (1m) can be used to find the angle of the ray with respect to the x-axis.

### Step 3: Calculate the Angle of the Ray
- The slope of the line connecting the origin (0,0) to the point (1, √3) can be calculated using the formula:
\[ \tan(\alpha) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3} \]
- Therefore, the angle α can be determined as:
\[ \alpha = \tan^{-1}(\sqrt{3}) = 60^\circ \]

### Step 4: Relate the Angles
- According to the law of reflection, the angle of incidence (i) is equal to the angle of reflection (r).
- If we consider the normal to the mirror at the point of incidence, the angle θ that the mirror makes with the x-axis can be related to α:
\[ \theta + \alpha = 90^\circ \]
- Since we found α to be 60°, we can substitute:
\[ \theta + 60^\circ = 90^\circ \]
- Solving for θ gives:
\[ \theta = 90^\circ - 60^\circ = 30^\circ \]

### Step 5: Conclusion
- Therefore, the angle of the plane mirror with the x-axis should be **30 degrees**.

### Final Answer:
The angle of the plane mirror with the x-axis should be **30 degrees**.

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