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

Options

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

Explanation

To find the individual focal lengths of a convex lens and a concave lens that are in contact, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- The ratio of the focal lengths of the convex lens (F1) and the concave lens (F2) is \( \frac{F1}{F2} = \frac{2}{3} \).
- The equivalent focal length (F) of the combination is 30 cm.

### Step 2: Express Focal Lengths in Terms of a Variable
From the ratio \( \frac{F1}{F2} = \frac{2}{3} \), we can express F1 and F2 in terms of a variable:
Let \( F1 = 2x \) and \( F2 = 3x \).

### Step 3: Use the Formula for Equivalent Focal Length
The formula for the equivalent focal length \( F \) of two lenses in contact is given by:
\[ \frac{1}{F} = \frac{1}{F1} + \frac{1}{F2} \]
Substituting the values we have:
\[ \frac{1}{30} = \frac{1}{2x} + \frac{1}{3x} \]

### Step 4: Find a Common Denominator and Simplify
To combine the fractions on the right side, we find a common denominator:
\[ \frac{1}{30} = \frac{3}{6x} + \frac{2}{6x} = \frac{5}{6x} \]
So, we have:
\[ \frac{1}{30} = \frac{5}{6x} \]

### Step 5: Cross-Multiply to Solve for x
Cross-multiplying gives:
\[ 6x = 150 \]
Thus,
\[ x = 25 \]

### Step 6: Calculate Individual Focal Lengths
Now, substituting back to find F1 and F2:
\[ F1 = 2x = 2(25) = 50 \text{ cm} \]
\[ F2 = 3x = 3(25) = 75 \text{ cm} \]

### Step 7: Adjust the Sign for the Concave Lens
Since F2 is the focal length of a concave lens, it will be negative:
\[ F2 = -75 \text{ cm} \]

### Final Answer
The individual focal lengths are:
- Focal length of the convex lens (F1) = 50 cm
- Focal length of the concave lens (F2) = -75 cm