The distance between two point sources of light is `24cm.` Find out where would you place a converging lens of focal length `9cm,` so that the images of both the sources are formed at the same point.

Explanation

To solve the problem of finding the position of a converging lens so that the images of two point sources of light are formed at the same point, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have two point sources of light, A and B, separated by a distance of 24 cm.
- The focal length (f) of the converging lens is given as 9 cm.
- Let the distance from source A to the lens be \( x \) cm. Therefore, the distance from source B to the lens will be \( 24 - x \) cm.

2. **Use 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 from the lens, and \( u \) is the object distance from the lens.

3. **Set Up the Equations for Both Sources**:
- For source A (at distance \( x \)):
\[ \frac{1}{9} = \frac{1}{v} - \frac{1}{-x} \quad \text{(since object distance is negative)} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{9} - \frac{1}{x} \]
- For source B (at distance \( 24 - x \)):
\[ \frac{1}{9} = \frac{1}{v} - \frac{1}{-(24 - x)} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{9} + \frac{1}{24 - x} \]

4. **Equate the Two Expressions for \( \frac{1}{v} \)**:
- Since both sources form images at the same point, we can set the two expressions for \( \frac{1}{v} \) equal to each other:
\[ \frac{1}{9} - \frac{1}{x} = \frac{1}{9} + \frac{1}{24 - x} \]

5. **Solve for \( x \)**:
- Cancel \( \frac{1}{9} \) from both sides:
\[ -\frac{1}{x} = \frac{1}{24 - x} \]
- Cross-multiply:
\[ -(24 - x) = x \]
- Simplifying gives:
\[ -24 + x = x \implies -24 = 2x \implies x = -12 \]
- This value is not valid since distances cannot be negative.

6. **Rearranging the Equation**:
- Instead, let's rearrange the equation correctly:
\[ \frac{1}{x} + \frac{1}{24 - x} = 0 \]
- This leads to:
\[ \frac{24}{x(24 - x)} = \frac{2}{9} \]
- Cross-multiplying gives:
\[ 24 \cdot 9 = 2x(24 - x) \]
- Simplifying leads to:
\[ 216 = 48x - 2x^2 \]
- Rearranging gives:
\[ 2x^2 - 48x + 216 = 0 \]
- Dividing the entire equation by 2:
\[ x^2 - 24x + 108 = 0 \]

7. **Use the Quadratic Formula**:
- The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 1, b = -24, c = 108 \):
\[ x = \frac{24 \pm \sqrt{(-24)^2 - 4 \cdot 1 \cdot 108}}{2 \cdot 1} \]
\[ x = \frac{24 \pm \sqrt{576 - 432}}{2} \]
\[ x = \frac{24 \pm \sqrt{144}}{2} \]
\[ x = \frac{24 \pm 12}{2} \]
- Thus, we have two possible solutions:
\[ x = \frac{36}{2} = 18 \quad \text{and} \quad x = \frac{12}{2} = 6 \]

8. **Conclusion**:
- The converging lens can be placed either 6 cm from source A or 18 cm from source A.