Let `alpha_1,alpha_2` and `beta_1, beta_2` be the roots of the equation `ax^2+bx+c=0` and `px^2+qx+r=0` respectively. If the system of equations `alpha_1y+alpha_2z=0` and `beta_1y+beta_2z=0` has a non trivial solution then prove that `b^2/q^2=(ac)/(pr)`

Explanation

To prove that \( \frac{b^2}{q^2} = \frac{ac}{pr} \) given the conditions in the problem, we can follow these steps:

### Step 1: Identify the roots of the equations
The roots of the quadratic equations are given by:
- For \( ax^2 + bx + c = 0 \):
- Roots: \( \alpha_1, \alpha_2 \)
- Sum of roots: \( \alpha_1 + \alpha_2 = -\frac{b}{a} \)
- Product of roots: \( \alpha_1 \alpha_2 = \frac{c}{a} \)
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