A spherical mirror forms an image of magnification3. The object distance, if focal length of mirror is 24cm, may be

Options

A. 32cm, 24cm
B. 32cm, 16cm
C. 32cm only
D. 16cm only

Explanation

To solve the problem, we need to find the object distance (u) given the magnification (m) and the focal length (f) of a spherical mirror. The magnification is given as 3, and the focal length is -24 cm (negative for a concave mirror).

### Step-by-Step Solution:

1. **Understand the Magnification Formula**:
The magnification (m) is defined as the ratio of the image distance (v) to the object distance (u):
\[ m = \frac{v}{u} \]
We also know that for mirrors, magnification can be expressed in terms of focal length (f) and object distance (u):
\[ m = \frac{f}{f - u} \]

2. **Set Up the Equation**:
Since we know that \( m = 3 \) and \( f = -24 \) cm, we can set up the equation:
\[ 3 = \frac{-24}{-24 - u} \]

3. **Cross Multiply**:
To eliminate the fraction, we can cross-multiply:
\[ 3(-24 - u) = -24 \]

4. **Distribute the 3**:
Distributing the 3 gives:
\[ -72 - 3u = -24 \]

5. **Rearrange the Equation**:
Now, we can rearrange the equation to isolate \( u \):
\[ -3u = -24 + 72 \]
\[ -3u = 48 \]

6. **Solve for u**:
Dividing both sides by -3 gives:
\[ u = -16 \text{ cm} \]

7. **Check for the Second Possible Value**:
Since magnification can also be negative, we check for \( m = -3 \):
\[ -3 = \frac{-24}{-24 - u} \]
Cross-multiplying again:
\[ -3(-24 - u) = -24 \]
This simplifies to:
\[ 72 + 3u = -24 \]
Rearranging gives:
\[ 3u = -24 - 72 \]
\[ 3u = -96 \]
Dividing by 3:
\[ u = -32 \text{ cm} \]

### Final Answers:
The possible object distances are:
- \( u = -16 \text{ cm} \)
- \( u = -32 \text{ cm} \)