A concave lens is placed in contact with a convex lens of focal length `25 cm`. The combination produces a real image at a distance of `80 cm`, when an object is at a distance of `40 cm`. What is the focal length of concave lens ?

Options

A. `-400 cm`
B. `-200 cm`
C. `+400 cm`
D. `+200 cm`

Explanation

To find the focal length of the concave lens in the given problem, we can follow these steps:

### Step 1: Understand the given information
- Focal length of the convex lens, \( f_2 = +25 \, \text{cm} \)
- Object distance, \( u = -40 \, \text{cm} \) (negative as per sign convention)
- Image distance, \( v = +80 \, \text{cm} \) (positive as it is a real image)

### Step 2: Use the lens formula to find the focal length of the combination
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{f} = \frac{1}{80} - \left(-\frac{1}{40}\right) \]
This simplifies to:
\[ \frac{1}{f} = \frac{1}{80} + \frac{1}{40} \]

### Step 3: Find a common denominator and calculate
The common denominator for \( 80 \) and \( 40 \) is \( 80 \):
\[ \frac{1}{f} = \frac{1}{80} + \frac{2}{80} = \frac{3}{80} \]
Thus, the focal length of the combination is:
\[ f = \frac{80}{3} \approx 26.67 \, \text{cm} \]

### Step 4: Use the formula for the combination of lenses
The formula for the combination of two lenses in contact is:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \]
Where \( f_1 \) is the focal length of the concave lens and \( f_2 \) is the focal length of the convex lens.

### Step 5: Rearranging the equation to find \( f_1 \)
Rearranging gives:
\[ \frac{1}{f_1} = \frac{1}{f} - \frac{1}{f_2} \]
Substituting the known values:
\[ \frac{1}{f_1} = \frac{3}{80} - \frac{1}{25} \]

### Step 6: Find a common denominator and calculate
The common denominator for \( 80 \) and \( 25 \) is \( 200 \):
\[ \frac{1}{f_1} = \frac{3 \times 2.5}{200} - \frac{8}{200} = \frac{7.5 - 8}{200} = \frac{-0.5}{200} \]
Thus:
\[ f_1 = -400 \, \text{cm} \]

### Conclusion
The focal length of the concave lens is \( -400 \, \text{cm} \).