A thin convex lens of focal length `25 cm` is cut into two pieces `0.5 cm` above the principal axis. The top part is placed at (0,0) and an object placed at `(-50 cm, 0)`. Find the coordinates of the image.

Options

A. `(50cm, -2 cm)`
B. `(50 cm, -1 cm)`
C. `(3 cm, -50 cm)`
D. `(60 cm, -25 cm)`

Explanation

To find the coordinates of the image formed by a thin convex lens that has been cut and repositioned, we can follow these steps:

### Step 1: Understand the Setup
We have a thin convex lens with a focal length \( f = 25 \, \text{cm} \). The lens is cut into two pieces, and the top part is positioned \( 0.5 \, \text{cm} \) above the principal axis. The object is placed at coordinates \( (-50 \, \text{cm}, 0) \).

### Step 2: Identify Object Distance
The object distance \( u \) is measured from the lens to the object. Since the object is placed at \( (-50 \, \text{cm}) \), we have:
\[ u = -50 \, \text{cm} \]
(Note: The negative sign indicates that the object is on the same side as the incoming light.)

### Step 3: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]
Substituting \( f = 25 \, \text{cm} \) and \( u = -50 \, \text{cm} \):
\[ \frac{1}{v} = \frac{1}{25} + \frac{1}{-50} \]
Calculating the right-hand side:
\[ \frac{1}{v} = \frac{2}{50} - \frac{1}{50} = \frac{1}{50} \]
Thus,
\[ v = 50 \, \text{cm} \]

### Step 4: Determine the Magnification
The magnification \( m \) is given by:
\[ m = \frac{v}{u} \]
Substituting the values:
\[ m = \frac{50}{-50} = -1 \]
The negative sign indicates that the image is inverted.

### Step 5: Calculate the Height of the Image
If the height of the object \( h \) is \( 0.5 \, \text{cm} \), then the height of the image \( h' \) can be calculated using:
\[ m = \frac{h'}{h} \]
Substituting the values:
\[ -1 = \frac{h'}{0.5} \]
This gives:
\[ h' = -0.5 \, \text{cm} \]
The negative sign indicates that the image is inverted.

### Step 6: Determine the Coordinates of the Image
The image is formed \( 50 \, \text{cm} \) from the lens along the principal axis and \( 0.5 \, \text{cm} \) below the principal axis. Thus, the coordinates of the image are:
\[ (50 \, \text{cm}, -0.5 \, \text{cm}) \]

### Final Answer
The coordinates of the image are:
\[ (50 \, \text{cm}, -1 \, \text{cm}) \]