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 $\frac{b^2}{b^2_1}=\frac{ac}{a_1 c_1}$?

Let the roots of the first quadratic be $\alpha$ and $t\alpha$ and the roots of the second be $\beta$ and $t\beta$
Then $$\alpha(t+1)=\frac ca$$ and $$\alpha^2t=\frac ca$$
Then $$\frac {b^2}{a^2}=\alpha(t+1)^2=\frac{c}{at}(t+1)^2$$
In exactly the same way,  $$\frac {b_1^2}{a_1^2}=\beta(t+1)^2=\frac{c_1}{a_1t}(t+1)^2$$
Now eliminate all the $t$ terms by division and the result follows immediately
