The focal length of convex tens is `f` and the distance of an object from the principal focus is x. The ratio of the size of the real image to the size of the object is

Options

A. `(f)/(x)`
B. `(x)/(f)`
C. `(f+x)/(f)`
D. `(f)/(f+x)`

Explanation

To find the ratio of the size of the real image to the size of the object for a convex lens, we can use the lens formula and magnification formula. Here’s a step-by-step solution:

### Step 1: Understand the Lens Formula
The lens formula for a convex lens is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) is the focal length of the lens,
- \( v \) is the image distance from the lens,
- \( u \) is the object distance from the lens.

### Step 2: Define Object Distance
Given that the distance of the object from the principal focus is \( x \), we can express the object distance \( u \) as:
\[ u = - (f - x) \]
This is because the object is located at a distance \( x \) from the focal point, and we take the convention that distances measured against the direction of the incoming light are negative.

### Step 3: Substitute into the Lens Formula
Now, substituting \( u \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{-(f - x)} \]
This simplifies to:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{(f - x)} \]

### Step 4: Rearranging the Equation
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{(f - x)} \]
Finding a common denominator:
\[ \frac{1}{v} = \frac{(f - x) - f}{f(f - x)} = \frac{-x}{f(f - x)} \]
Thus, we can express \( v \) as:
\[ v = -\frac{f(f - x)}{x} \]

### Step 5: Calculate Magnification
The magnification \( m \) of the lens is given by:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
Substituting the values of \( v \) and \( u \):
\[ m = -\left(-\frac{f(f - x)}{x}\right) \div \left(- (f - x)\right) \]
This simplifies to:
\[ m = \frac{f(f - x)}{x(f - x)} = \frac{f}{x} \]

### Step 6: Ratio of Sizes
The ratio of the size of the real image to the size of the object is therefore:
\[ \frac{h'}{h} = \frac{f}{x} \]

### Final Answer
The ratio of the size of the real image to the size of the object is:
\[ \frac{h'}{h} = \frac{f}{x} \]

---