how to prove $\frac{b^2}{b_1^2}=\frac{ac}{a_{1c_1}}$? If the Ratio of the roots of $ax^2+bx+c=0$ be equal to the ratio of the roots of $a_1x^2+b_1x+c_1=0$, then how one prove that $\frac{b^2}{b^2_1}=\frac{ac}{a_1 c_1}$?

Hint :
let $\alpha$ and $\beta$ be the roots of $ax^2+bx+c=0$ & let $\gamma$ and $\delta$ be the roots of $a_1 x^2+b_1 x+c_1 =0$.
The ratio of their roots are equal if 
\begin{eqnarray*}
\frac{\alpha}{\beta} = \frac{\gamma}{\delta}.
\end{eqnarray*}
Further hint : $\color{red}{\alpha+\beta=-\frac{b}{a}}$ & $\alpha \beta=\frac{c}{a}$
\begin{eqnarray*}
\frac{b^2}{ac} = \color{red}{\frac{b^2}{a^2}} \frac{a}{c} = \frac{\color{red}{(\alpha+\beta)^2}}{\alpha \beta}=\frac{\alpha}{\beta}+2+\frac{\beta}{\alpha} = \cdots
\end{eqnarray*}
