Prove common ratio of GP from a equation

Question

If $(a²+b²)x²-2b(c+a)(b/a)x+(b²+c²)=0$ has equal roots, how do I prove that $a$, $b$, $c$ are in G.P. and their common ratio is $x$.

Since the roots are equal

$D=b²-4ac=0$

I've put the values and I get

$b=√(ac)$

which means that $a$, $b$, $c$ are in G.P. But how do prove that $x$ is the common ratio?

Answer 1

Let r be the common ratio. Then, $r^2 = \dfrac ca$.

Also, if $\alpha$ is a root, then $\alpha^2 =$ the product of the roots $= \dfrac {b^2 + c^2}{a^2 + b^2}$. The last fraction can be converted to $\dfrac ca$ through substitution of $b^2$ by $ca$ and cancellations.

Therefore, $r^2 = \alpha^2$.