An object is kept at a distance of 4 cm form the first focus of a convex lens. A real image is formed at a distance of 9 cm from its second focus. What is the focal length of that lens is ___________(in cm)

Answer

To find the focal length of the convex lens, we can follow these steps:

### Step 1: Understand the problem
We know that the object is placed at a distance of 4 cm from the first focus of the lens. The image is formed at a distance of 9 cm from the second focus of the lens.

### Step 2: Define the variables
Let:
- \( f \) = focal length of the lens (in cm)
- The distance of the object from the first focus = 4 cm.
- The object distance \( u \) can be expressed as \( u = - (f - 4) \).
- The distance of the image from the second focus = 9 cm.
- The image distance \( v \) can be expressed as \( v = f + 9 \).

### Step 3: Write the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

### Step 4: Substitute the values of \( u \) and \( v \)
Substituting the expressions for \( u \) and \( v \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{f + 9} - \frac{1}{-(f - 4)} \]

### Step 5: Simplify the equation
This can be rewritten as:
\[ \frac{1}{f} = \frac{1}{f + 9} + \frac{1}{f - 4} \]

### Step 6: Find a common denominator
The common denominator for the right side is \((f + 9)(f - 4)\):
\[ \frac{1}{f} = \frac{(f - 4) + (f + 9)}{(f + 9)(f - 4)} \]
\[ \frac{1}{f} = \frac{2f + 5}{(f + 9)(f - 4)} \]

### Step 7: Cross-multiply
Cross-multiplying gives:
\[ (f + 9)(f - 4) = f(2f + 5) \]

### Step 8: Expand both sides
Expanding both sides:
\[ f^2 + 5f - 36 = 2f^2 + 5f \]

### Step 9: Rearrange the equation
Rearranging gives:
\[ 0 = 2f^2 - f^2 + 5f - 5f + 36 \]
\[ 0 = f^2 + 36 \]

### Step 10: Solve for \( f \)
This simplifies to:
\[ f^2 = 36 \]
Taking the square root:
\[ f = 6 \text{ cm} \]

### Final Answer
The focal length of the lens is \( 6 \) cm.

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