Where should a convex lens of focal length 9cm be placed (in cm) between two point sources `S_(1)` and `S_(2)` where are 24 cm apart, so that images of both the sources are formed at the same place. You have to find distance of lens from `S_(1)` or `S_(2)` whichever is lesser.

Explanation

To solve the problem of where to place a convex lens of focal length 9 cm between two point sources \( S_1 \) and \( S_2 \) that are 24 cm apart, we will use the lens formula and the concept of image formation.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the distance of the lens from \( S_1 \) be \( x \).
- Therefore, the distance of the lens from \( S_2 \) will be \( 24 - x \).
- The focal length \( f \) of the convex lens is given as 9 cm.

2. **Apply the Lens Formula:**
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance.

3. **Set Up the Equations for Both Sources:**
- For \( S_1 \):
- The object distance \( u_1 = -x \) (negative because the object is on the same side as the incoming light).
- Let the image distance for \( S_1 \) be \( y \).
- The lens formula becomes:
\[ \frac{1}{9} = \frac{1}{y} - \frac{1}{-x} \quad \text{(Equation 1)} \]
Rearranging gives:
\[ \frac{1}{y} = \frac{1}{9} - \frac{1}{x} \]

- For \( S_2 \):
- The object distance \( u_2 = -(24 - x) \).
- Let the image distance for \( S_2 \) also be \( y \).
- The lens formula becomes:
\[ \frac{1}{9} = \frac{1}{y} - \frac{1}{-(24 - x)} \quad \text{(Equation 2)} \]
Rearranging gives:
\[ \frac{1}{y} = \frac{1}{9} + \frac{1}{(24 - x)} \]

4. **Equate the Two Expressions for \( \frac{1}{y} \):**
From Equation 1 and Equation 2, we have:
\[ \frac{1}{9} - \frac{1}{x} = \frac{1}{9} + \frac{1}{(24 - x)} \]
Simplifying this gives:
\[ -\frac{1}{x} = \frac{1}{(24 - x)} \]
Cross-multiplying leads to:
\[ - (24 - x) = x \]
This simplifies to:
\[ -24 + x = x \]
Thus:
\[ 24 = 2x \quad \Rightarrow \quad x = 12 \text{ cm} \]

5. **Finding the Distances:**
Since \( x = 12 \) cm, the distance from \( S_1 \) to the lens is 12 cm, and the distance from \( S_2 \) to the lens is \( 24 - 12 = 12 \) cm. Therefore, the distance of the lens from either source is the same.

### Conclusion:
The lens should be placed 12 cm from either point source \( S_1 \) or \( S_2 \).