Prove that for spherical mirrors the product of the distance of the object and the image to the
principal focus is always equal to the square of the principal focal length.

Explanation

To prove that for spherical mirrors, the product of the distance of the object and the image to the principal focus is always equal to the square of the principal focal length, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( f \) = focal length of the mirror
- \( u \) = object distance from the pole of the mirror
- \( v \) = image distance from the pole of the mirror

### Step 2: Determine Distances from the Focus
The distance of the object from the principal focus is given by:
\[ u - f \]
The distance of the image from the principal focus is given by:
\[ v - f \]

### Step 3: Write the Product of Distances
Now, we can write the product of the distances from the focus:
\[ (u - f)(v - f) \]

### Step 4: Expand the Product
Expanding the product gives:
\[ (u - f)(v - f) = uv - uf - vf + f^2 \]

### Step 5: Use the Mirror Formula
From the mirror formula, we know:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Rearranging this gives:
\[ \frac{u + v}{uv} = \frac{1}{f} \]
Multiplying both sides by \( uv \) gives:
\[ u + v = \frac{uv}{f} \]

### Step 6: Substitute \( u + v \) into the Expanded Product
Now, substituting \( u + v \) into our expanded product:
\[ uv - f(u + v) + f^2 = uv - f\left(\frac{uv}{f}\right) + f^2 \]
This simplifies to:
\[ uv - uv + f^2 = f^2 \]

### Step 7: Conclusion
Thus, we have shown that:
\[ (u - f)(v - f) = f^2 \]
This proves that the product of the distance of the object and the image to the principal focus is equal to the square of the principal focal length.