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

Options

A. `-75, 50`
B. `-10, 15`
C. `-75, 25`
D. `-15, 10`

Answer

To find the individual focal lengths of the convex and concave lenses in contact, we can follow these steps:

### Step 1: Define the focal lengths
Let the focal length of the convex lens be \( f \) (positive value) and the focal length of the concave lens be \( -\frac{3}{2}f \) (negative value, since it is a concave lens).

### Step 2: Use the formula for equivalent focal length
The formula for the equivalent focal length \( F \) of two lenses in contact is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
In our case:
\[ \frac{1}{30} = \frac{1}{f} + \frac{1}{-\frac{3}{2}f} \]

### Step 3: Substitute the values
Substituting the values into the equation:
\[ \frac{1}{30} = \frac{1}{f} - \frac{2}{3f} \]

### Step 4: Simplify the equation
To simplify, find a common denominator:
\[ \frac{1}{30} = \frac{3}{3f} - \frac{2}{3f} = \frac{1}{3f} \]

### Step 5: Cross-multiply to solve for \( f \)
Cross-multiplying gives:
\[ 1 \cdot 3f = 30 \cdot 1 \]
\[ 3f = 30 \]
\[ f = 10 \text{ cm} \]

### Step 6: Find the focal length of the concave lens
Now, substitute \( f \) back to find the focal length of the concave lens:
\[ f_{concave} = -\frac{3}{2}f = -\frac{3}{2} \times 10 = -15 \text{ cm} \]

### Conclusion
The individual focal lengths are:
- Focal length of the convex lens: \( 10 \text{ cm} \)
- Focal length of the concave lens: \( -15 \text{ cm} \)