If three unequel numbers are in HP and their squares are in AP, show that they are in the ratio `1+sqrt(3):-2:1-sqrt(3) " or " 1-sqrt(3):-2:1+sqrt(3)`.

Explanation

To solve the problem, we need to show that if three unequal numbers \( p, q, r \) are in Harmonic Progression (HP) and their squares \( p^2, q^2, r^2 \) are in Arithmetic Progression (AP), then they are in the ratio \( 1 + \sqrt{3} : -2 : 1 - \sqrt{3} \) or \( 1 - \sqrt{3} : -2 : 1 + \sqrt{3} \).

### Step-by-Step Solution:

1. **Understanding HP and AP**:
- If \( p, q, r \) are in HP, then the reciprocals \( \frac{1}{p}, \frac{1}{q}, \frac{1}{r} \) are in AP. This means:
\[
\frac{2}{q} = \frac{1}{p} + \frac{1}{r}
...