Two lenses of focal lengths `20 cm and - 40 cm` are held in contact. The image of an object at infinity will be formed by the combination at

Options

A. `10 cm`
B. `20 cm`
C. 40 cm
D. infinity

Explanation

To solve the problem of finding the image distance formed by a combination of two lenses with focal lengths of \(20 \, \text{cm}\) (convex lens) and \(-40 \, \text{cm}\) (concave lens) when an object is placed at infinity, we can follow these steps:

### Step 1: Identify the focal lengths
- The focal length of the first lens (convex) is \(f_1 = 20 \, \text{cm}\).
- The focal length of the second lens (concave) is \(f_2 = -40 \, \text{cm}\).

### Step 2: Calculate the equivalent focal length of the lens combination
The formula for the equivalent focal length \(f\) of two lenses in contact is given by:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \]
Substituting the values:
\[ \frac{1}{f} = \frac{1}{20} + \frac{1}{-40} \]

### Step 3: Simplify the equation
To simplify, we find a common denominator:
\[ \frac{1}{f} = \frac{2}{40} - \frac{1}{40} = \frac{1}{40} \]

### Step 4: Calculate the equivalent focal length
Now, taking the reciprocal gives:
\[ f = 40 \, \text{cm} \]

### Step 5: Use the lens formula to find the image distance \(V\)
The lens formula is given by:
\[ \frac{1}{V} - \frac{1}{U} = \frac{1}{f} \]
Since the object is at infinity, \(U = -\infty\), thus:
\[ \frac{1}{V} - 0 = \frac{1}{40} \]
This simplifies to:
\[ \frac{1}{V} = \frac{1}{40} \]

### Step 6: Calculate the image distance \(V\)
Taking the reciprocal gives:
\[ V = 40 \, \text{cm} \]

### Conclusion
The image of an object at infinity will be formed by the combination of the two lenses at a distance of \(40 \, \text{cm}\) from the lens combination.

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