Let `a, b, c, d` be real numbers in `G.P.` If `u, v, w` satisfy the system of equations `u + 2y +3w = 6,4u + 5v + 6w =12 and 6u + 9v = 4` then show that the roots of the equation `(1/u+1/v+/w)x^2+[(b-c)^2+(c-a)^2+(d-b)^2]x+u+v+w=0` and 20x^2+10(a-d)^2 x-9=0` are reciprocals of each other.

Options

A. `a-d`
B. `(a-d)^(2)`
C. `a^(2)-d^(2)`
D. `(a+d)^(2)`

Explanation

To solve the given problem step by step, we will first solve the system of equations for \( u, v, w \), then we will calculate the expression involving \( b, c, a, d \), and finally show that the roots of the two equations are reciprocals of each other.

### Step 1: Solve the System of Equations

We have the following equations:

1. \( u + 2v + 3w = 6 \) (Equation 1)
2. \( 4u + 5v + 6w = 12 \) (Equation 2)
...