The object distance u, the image distance v and the magnification m in a lens follow certain linear relations.

Options

A. `(1)/(u) "versus"(1)/(v)`
B. m versus u
C. u versus v
D. m versus v

Answer

To solve the problem regarding the linear relations among the object distance \( u \), the image distance \( v \), and the magnification \( m \) in a lens, we can follow these steps:

### Step 1: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Rearranging this gives:
\[ \frac{1}{v} = \frac{1}{u} + \frac{1}{f} \]
This can be considered as a linear equation in the form of \( y = mx + c \), where:
- \( y = \frac{1}{v} \)
- \( x = \frac{1}{u} \)
- \( m = 1 \) (the slope)
- \( c = \frac{1}{f} \) (the y-intercept)

### Step 2: Use the Magnification Formula
The magnification \( m \) is defined as:
\[ m = -\frac{v}{u} \]
This can be rearranged to express \( v \) in terms of \( m \) and \( u \):
\[ v = -mu \]
This can also be rearranged to fit a linear equation format:
\[ m = -\frac{1}{u} v \]
This can be considered as another linear equation where:
- \( y = m \)
- \( x = v \)
- \( m = -\frac{1}{u} \) (the slope)

### Step 3: Identify the Relationships
From the above equations, we can identify two linear relationships:
1. The relationship between \( \frac{1}{v} \) and \( \frac{1}{u} \).
2. The relationship between \( m \) and \( v \).

### Step 4: Conclusion
Thus, we can conclude that the object distance \( u \), the image distance \( v \), and the magnification \( m \) in a lens follow certain linear relations as described above.