A convex lens is in contact with a concave lens. The magnitude of the ration of their focal lengths is `2//3`. Their equivalent focal length is 30cm. What are their individual focal lengths?

Options

A. `-15,10`
B. `-10,15`
C. `75,50`
D. `-75,50`

Explanation

To solve the problem, we need to find the individual focal lengths of a convex lens (f1) and a concave lens (f2) that are in contact, given the ratio of their focal lengths and their equivalent focal length.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The ratio of the focal lengths of the convex lens (f1) and the concave lens (f2) is given as \( \frac{f1}{f2} = \frac{2}{3} \).
- The equivalent focal length (F) of the combination is given as \( F = 30 \, \text{cm} \).

2. **Expressing the Focal Lengths in Terms of Each Other**:
- From the ratio \( \frac{f1}{f2} = \frac{2}{3} \), we can express \( f2 \) in terms of \( f1 \):
\[ f2 = \frac{3}{2} f1 \]

3. **Using the Formula for Equivalent Focal Length**:
- For two lenses in contact, the formula for the equivalent focal length is:
\[ \frac{1}{F} = \frac{1}{f1} + \frac{1}{f2} \]
- Substituting \( f2 \) from the previous step into this equation gives:
\[ \frac{1}{30} = \frac{1}{f1} + \frac{1}{\left(\frac{3}{2} f1\right)} \]

4. **Simplifying the Equation**:
- The term \( \frac{1}{\left(\frac{3}{2} f1\right)} \) can be rewritten as \( \frac{2}{3 f1} \):
\[ \frac{1}{30} = \frac{1}{f1} + \frac{2}{3 f1} \]
- Combining the fractions on the right-hand side:
\[ \frac{1}{30} = \frac{3 + 2}{3 f1} = \frac{5}{3 f1} \]

5. **Cross-Multiplying to Solve for f1**:
- Cross-multiplying gives:
\[ 1 \cdot 3 f1 = 30 \cdot 5 \]
\[ 3 f1 = 150 \]
\[ f1 = \frac{150}{3} = 50 \, \text{cm} \]

6. **Finding f2**:
- Now substituting \( f1 \) back into the equation for \( f2 \):
\[ f2 = \frac{3}{2} f1 = \frac{3}{2} \cdot 50 = 75 \, \text{cm} \]

7. **Considering the Sign of f2**:
- Since f2 is a concave lens, it will be negative:
\[ f2 = -75 \, \text{cm} \]

### Final Result:
- The individual focal lengths are:
- \( f1 = 50 \, \text{cm} \) (Convex lens)
- \( f2 = -75 \, \text{cm} \) (Concave lens)