https://file.aiquickdraw.com/mathbot/file/1725061534519l44oy8ia.jpg

To find \( U \) and \( V \), let's analyze the equations shown in the image.

1. The first equation appears to be the lens equation:
   \[
   \frac{1}{F} = \frac{1}{U} + \frac{1}{V}
   \]
   Where \( F \) is the focal length, \( U \) is the object distance, and \( V \) is the image distance.

2. Given values:
   - \( F = 10 \) (it seems to be focal length in some units)
   - \( M = 0.25 \) (Magnification, where \( M = \frac{V}{U} \))

3. The relationship between magnification \( M \), object distance \( U \), and image distance \( V \) is:
   \[
   M = \frac{V}{U}
   \]

### Step 1: Express \( V \) in terms of \( U \)
Since \( M = 0.25 \), we have:
\[
V = 0.25U
\]

### Step 2: Substitute \( V = 0.25U \) into the lens equation
The lens equation is:
\[
\frac{1}{F} = \frac{1}{U} + \frac{1}{V}
\]
Substitute \( F = 10 \) and \( V = 0.25U \):
\[
\frac{1}{10} = \frac{1}{U} + \frac{1}{0.25U}
\]

### Step 3: Simplify the equation
First, simplify \( \frac{1}{0.25U} \) to \( \frac{4}{U} \), so the equation becomes:
\[
\frac{1}{10} = \frac{1}{U} + \frac{4}{U}
\]
\[
\frac{1}{10} = \frac{5}{U}
\]

### Step 4: Solve for \( U \)
\[
U = 10 \times 5 = 50
\]

### Step 5: Find \( V \)
Now that we have \( U = 50 \), use the magnification equation \( V = 0.25U \) to find \( V \):
\[
V = 0.25 \times 50 = 12.5
\]

### Final Answer
- \( U = 50 \) units
- \( V = 12.5 \) units

Would you like further details or explanations? Here are some follow-up questions you might find useful:

1. How does the lens equation change for a concave lens?
2. What happens if the magnification is greater than 1?
3. How do you derive the lens equation from first principles?
4. What is the significance of the sign convention in lens equations?
5. How does the magnification formula relate to the size of the image?

**Tip:** When using the lens formula, always be cautious about the sign convention (whether the lens is converging or diverging) as it can affect the results.

Learn how to find the object distance \( U \) and image distance \( V \) using the lens equation and magnification formula. Step-by-step solution provided with detailed calculations. Suitable for advanced high school students studying optics and lens equations.

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