A light ray travelling parallel to the principal axis of a concave mirror strikes the mirror at angle of incidence `theta`. If radius of curvature is `R`, then after reflection, the ray meets the principal axis at a distance d from the centre of curvature then d is

Options

A. `(R)/(2)`
B. `(Rcostheta)/(2)`
C. `(R)/(2costheta)`
D. `(R)/(2)(1+costheta)`

Explanation

To solve the problem step by step, we will analyze the situation involving a concave mirror and the light ray striking it.

### Step 1: Understand the Geometry
A light ray is traveling parallel to the principal axis of a concave mirror and strikes the mirror at an angle of incidence \( \theta \). The radius of curvature of the mirror is \( R \). The center of curvature (C) is located at a distance \( R/2 \) from the mirror's surface.

**Hint:** Visualize the setup with a diagram showing the concave mirror, the principal axis, the incident ray, and the reflected ray.

### Step 2: Identify Angles
When the light ray strikes the mirror, the angle of incidence \( \theta \) is equal to the angle of reflection \( \theta \) due to the law of reflection. The normal to the surface at the point of incidence is perpendicular to the mirror surface.

**Hint:** Remember that the angle of incidence equals the angle of reflection.

### Step 3: Analyze the Triangle Formed
The triangle formed by the incident ray, the reflected ray, and the line connecting the point of incidence to the center of curvature can be analyzed. Let’s denote the points:
- A: Point where the ray strikes the mirror.
- B: The point where the reflected ray meets the principal axis.
- C: The center of curvature.

In triangle ABC:
- Angle A is \( \theta \) (angle of incidence).
- Angle B is \( \theta \) (angle of reflection).
- Angle C is \( \pi - 2\theta \) (since the sum of angles in a triangle is \( \pi \)).

**Hint:** Use the properties of triangles to relate the angles and sides.

### Step 4: Apply the Sine Rule
Using the sine rule in triangle ABC:
\[ \frac{d}{\sin(\pi - 2\theta)} = \frac{R}{\sin(\theta)} \]
Since \( \sin(\pi - x) = \sin(x) \), we can simplify:
\[ \frac{d}{\sin(2\theta)} = \frac{R}{\sin(\theta)} \]

**Hint:** Recall the sine rule and how to apply it to relate sides and angles in a triangle.

### Step 5: Solve for \( d \)
From the sine rule, we can express \( d \):
\[ d = \frac{R \cdot \sin(2\theta)}{\sin(\theta)} \]
Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we substitute:
\[ d = \frac{R \cdot 2 \sin(\theta) \cos(\theta)}{\sin(\theta)} = 2R \cos(\theta) \]

### Step 6: Final Expression for \( d \)
Since the radius of curvature \( R \) is related to the focal length \( f \) of the mirror by \( R = 2f \), we can express \( d \) as:
\[ d = \frac{R}{2} \cos(\theta) \]

### Conclusion
Thus, the distance \( d \) from the center of curvature to where the ray meets the principal axis is given by:
\[ d = \frac{R}{2} \cos(\theta) \]

**Final Answer:** \( d = \frac{R}{2} \cos(\theta) \)