Which of the following relations is correct for a spherical mirror if a point object is kept on the principal axis ['P' is pole 'C' is centre object is at point 'O'. Image is at point 'I'].

Options

A. `(OP)/(OC)=(IP)/(IC)`
B. `(OP)/(IC)=(IP)/(OC)`
C. `(PC)/(PO)=(PI)/(PC)`
D. `(IO)/(CP)=(IP)/(CO)`

Explanation

To solve the problem regarding the correct relation for a spherical mirror when a point object is placed on the principal axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a spherical mirror with a pole (P), center of curvature (C), an object located at point O, and an image formed at point I.

2. **Identify the Triangles**:
- We can form two triangles based on the positions of the object and the image. The first triangle is formed by points O, O' (the image of the object), and P (the pole of the mirror). The second triangle is formed by points I, I' (the image of the image), and P.

3. **Establish Similarity of Triangles**:
- Since both triangles share the angle at P and the angles at O and I are equal (due to the law of reflection), we can conclude that triangle OOP is similar to triangle IIP.

4. **Write the Ratio from Similar Triangles**:
- From the similarity of triangles, we can write the following relation:
\[ \frac{OO'}{II'} = \frac{OP}{IP} \]

5. **Consider the Other Set of Triangles**:
- Similarly, we can consider triangles OOC (where C is the center of curvature) and IIC. These triangles are also similar because they share the angle at C and have right angles at O and I.
- Thus, we can write:
\[ \frac{OO'}{II'} = \frac{OC}{IC} \]

6. **Combine the Relations**:
- Since both ratios are equal to \(\frac{OO'}{II'}\), we can equate them:
\[ \frac{OP}{IP} = \frac{OC}{IC} \]

7. **Final Relation**:
- Rearranging gives us:
\[ \frac{OP}{OC} = \frac{IC}{IP} \]
- This shows the relationship between the distances of the object and image from the pole and center of curvature.

### Conclusion:
The correct relation for a spherical mirror when a point object is kept on the principal axis is:
\[ \frac{OP}{OC} = \frac{IC}{IP} \]