An object is placed upright on the axis of a thin convex lens at a distance of four focal lengths (4f) from the center of the lens. An inverted image appears at a distance of `4//3` f on the other side of the lens. What is the ratio of the height of the image of the height of the object?

Options

A. `1//3`
B. `3//4`
C. `4//3`
D. `3//1`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the object distance (U) and image distance (V)
Given:
- The object is placed at a distance of 4f from the lens. Therefore, the object distance \( U = -4f \) (the negative sign indicates that the object is on the same side as the incoming light).
- The image is formed at a distance of \( \frac{4}{3}f \) on the other side of the lens. Therefore, the image distance \( V = +\frac{4}{3}f \) (the positive sign indicates that the image is on the opposite side of the lens).

### Step 2: Calculate the magnification (m)
The magnification \( m \) for a lens is given by the formula:
\[ m = \frac{V}{U} \]
Substituting the values of \( V \) and \( U \):
\[ m = \frac{\frac{4}{3}f}{-4f} \]
Simplifying this:
\[ m = \frac{4/3}{-4} = -\frac{1}{3} \]

### Step 3: Relate magnification to heights
The magnification is also defined as the ratio of the height of the image \( h_i \) to the height of the object \( h_o \):
\[ m = \frac{h_i}{h_o} \]
From our previous calculation, we have:
\[ -\frac{1}{3} = \frac{h_i}{h_o} \]
This implies:
\[ h_i = -\frac{1}{3} h_o \]

### Step 4: Find the ratio of heights
The ratio of the height of the image to the height of the object can be expressed as:
\[ \frac{h_i}{h_o} = -\frac{1}{3} \]
Thus, the ratio of the height of the image to the height of the object is:
\[ \frac{h_i}{h_o} = \frac{1}{3} \]
The negative sign indicates that the image is inverted.

### Final Answer
The ratio of the height of the image to the height of the object is \( \frac{1}{3} \).

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