Point A (0, 1 cm) and B (12 cm, 5 cm) are the coordinates of object and image x-axis is the principal axis of the mirror. The, this object image pair is

Options

A. due to a convex mirror of focal length 2.5 cm
B. due to concave mirror having its pole at (2 cm, 0)
C. due to a concave mirror having its pole at (-2 cm, 0)
D. Data is insufficient

Explanation

To solve the problem of identifying the type of mirror based on the given object and image coordinates, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Object A is at (0, 1 cm)
- Image B is at (12 cm, 5 cm)

2. **Determine the Object and Image Heights:**
- The height of the object (h_o) is 1 cm (positive, as it is above the principal axis).
- The height of the image (h_i) is 5 cm (also positive, indicating that the image is erect).

3. **Calculate Magnification (m):**
- Magnification (m) is given by the formula:
\[ m = \frac{h_i}{h_o} = \frac{5 \text{ cm}}{1 \text{ cm}} = 5 \]
- Since the image is erect and magnified, the magnification is positive.

4. **Use the Magnification Formula:**
- The magnification can also be expressed in terms of object distance (u) and image distance (v):
\[ m = -\frac{v}{u} \]
- Setting the two expressions for magnification equal gives:
\[ 5 = -\frac{v}{u} \]
- Rearranging gives:
\[ v = -5u \]

5. **Determine the Distance Between Object and Image:**
- The distance between the object and image is given as 12 cm. Therefore:
\[ |u| + |v| = 12 \]

6. **Substituting for v:**
- Substitute \( v = -5u \) into the distance equation:
\[ |u| + |-5u| = 12 \]
- This simplifies to:
\[ |u| + 5|u| = 12 \implies 6|u| = 12 \implies |u| = 2 \text{ cm} \]
- Thus, \( u = -2 \text{ cm} \) (the negative sign indicates that the object is in front of the mirror).

7. **Calculate v:**
- Using \( v = -5u \):
\[ v = -5(-2) = 10 \text{ cm} \]

8. **Determine the Focal Length (f):**
- Using the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substitute \( v = 10 \text{ cm} \) and \( u = -2 \text{ cm} \):
\[ \frac{1}{f} = \frac{1}{10} + \frac{1}{-2} = \frac{1}{10} - \frac{5}{10} = -\frac{4}{10} \]
- Therefore:
\[ f = -\frac{10}{4} = -2.5 \text{ cm} \]

9. **Conclusion on the Type of Mirror:**
- Since the focal length is negative, this indicates that the mirror is a **concave mirror**.

### Final Answer:
The object-image pair indicates that the mirror is a **concave mirror**.