We combined a convex lens of focal length `f_(1)` and concave lens of focal lengths `f_(2)` and their combined focal length was F . The combination of these lenses will behave like a concave lens, if

Options

A. `f_(1)gtf_(2)`
B. `f_(1)ltf_(2)`
C. `f_(1)=f_(2)`
D. `f_(1)lef_(2)`

Explanation

To determine the conditions under which the combination of a convex lens and a concave lens behaves like a concave lens, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Focal Lengths**:
- Let \( f_1 \) be the focal length of the convex lens (which is positive).
- Let \( f_2 \) be the focal length of the concave lens (which is negative).

2. **Power of Lenses**:
- The power \( P \) of a lens is given by the formula:
\[ P = \frac{1}{f} \]
- The power of the convex lens is \( P_1 = \frac{1}{f_1} \) (positive).
- The power of the concave lens is \( P_2 = \frac{1}{f_2} \) (negative).

3. **Combined Power**:
- The combined power \( P \) of the two lenses in contact is:
\[ P = P_1 + P_2 = \frac{1}{f_1} + \frac{1}{f_2} \]
- This can be rewritten as:
\[ P = \frac{f_2 + f_1}{f_1 f_2} \]

4. **Combined Focal Length**:
- The equivalent focal length \( F \) of the combination can be expressed as:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
- Rearranging gives:
\[ F = \frac{f_1 f_2}{f_1 + f_2} \]

5. **Condition for Concave Lens Behavior**:
- For the combination to behave like a concave lens, the equivalent focal length \( F \) must be negative:
\[ F < 0 \]
- Since \( f_1 \) is positive and \( f_2 \) is negative, we can analyze the expression:
\[ F = \frac{f_1 f_2}{f_1 + f_2} \]
- The numerator \( f_1 f_2 \) will be negative (since \( f_2 < 0 \)), and for \( F \) to be negative, the denominator \( f_1 + f_2 \) must also be positive.

6. **Analyzing the Denominator**:
- The condition for the denominator to be positive is:
\[ f_1 + f_2 > 0 \]
- Since \( f_2 \) is negative, we can rewrite this as:
\[ f_1 > |f_2| \]
- This means that the magnitude of the focal length of the convex lens must be greater than the magnitude of the focal length of the concave lens.

### Conclusion:
Thus, the combination of a convex lens of focal length \( f_1 \) and a concave lens of focal length \( f_2 \) will behave like a concave lens if:
\[ f_1 > |f_2| \]