A convex lens of focal length 20 cm produces images of the same magnification 2 when an object is kept at two distance `x_(1)` and `x_(2)(x_(1)gtx_(2))` from the lens. The ratio of `x_(1)` and `x_(2)` is

Options

A. `2:1`
B. `3:1`
C. `5:3`
D. `4:3`

Answer

To solve the problem, we need to find the ratio of the object distances \( x_1 \) and \( x_2 \) for a convex lens with a focal length of 20 cm, given that both distances produce images with the same magnification of 2.

### Step-by-Step Solution:

1. **Understand the Magnification Formula:**
The magnification \( m \) for a lens is given by:
\[ m = -\frac{b}{u} \]
where \( b \) is the image distance and \( u \) is the object distance.

2. **Set Up for \( x_1 \):**
For the first object distance \( x_1 \) (which produces a real image), we know:
\[ m = 2 \implies 2 = -\frac{b_1}{-x_1} \implies b_1 = 2x_1 \]

3. **Use the Lens Formula:**
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{b} - \frac{1}{u} \]
Substituting \( b_1 \) and \( u = -x_1 \):
\[ \frac{1}{20} = \frac{1}{2x_1} - \frac{1}{-x_1} \]
Simplifying this:
\[ \frac{1}{20} = \frac{1}{2x_1} + \frac{1}{x_1} \]
\[ \frac{1}{20} = \frac{1 + 2}{2x_1} = \frac{3}{2x_1} \]
Cross-multiplying gives:
\[ 3 = \frac{2x_1}{20} \implies x_1 = \frac{3 \times 20}{2} = 30 \text{ cm} \]

4. **Set Up for \( x_2 \):**
For the second object distance \( x_2 \) (which produces a virtual image), we again have:
\[ m = 2 \implies 2 = -\frac{b_2}{-x_2} \implies b_2 = 2x_2 \]

5. **Use the Lens Formula for \( x_2 \):**
Using the lens formula again:
\[ \frac{1}{20} = \frac{1}{2x_2} - \frac{1}{-x_2} \]
Simplifying this:
\[ \frac{1}{20} = \frac{1}{2x_2} + \frac{1}{x_2} \]
\[ \frac{1}{20} = \frac{1 + 2}{2x_2} = \frac{3}{2x_2} \]
Cross-multiplying gives:
\[ 3 = \frac{2x_2}{20} \implies x_2 = \frac{3 \times 20}{2} = 10 \text{ cm} \]

6. **Calculate the Ratio \( \frac{x_1}{x_2} \):**
Now we can find the ratio:
\[ \frac{x_1}{x_2} = \frac{30}{10} = 3 \]
Thus, the ratio \( x_1 : x_2 = 3 : 1 \).

### Final Answer:
The ratio of \( x_1 \) to \( x_2 \) is \( 3 : 1 \).