If r be the ratio of the roots of the equation ax^2+bx+c=0 show that (r+1)^2 / r = b^2 / ac

Solution:   Given equation is    $ax^{2}+bx+c=0 \hspace{1cm}............(1)$

Let   the roots of equation   (1)  be  $\alpha$  and  $r \alpha$ then

$\\ \\ \Rightarrow \hspace{1cm} \alpha+r\alpha =-\frac{b}{a}\hspace{1cm}.........(2)\\ \\ \Rightarrow \hspace{1cm}r\alpha^{2}=\frac{c}{a}\hspace{1cm}..........(3)\\ \\ \Rightarrow \hspace{1cm}\alpha=-\frac{b}{a(r+1)}$

Putting  the value of   $\alpha$ in $(3)$  , we get

$\\ \\ \Rightarrow \hspace{1cm}\frac{rb^{2}}{a^{2}(r+1)^{2}}=\frac{c}{a}\Rightarrow \frac{b^{2}}{ac}=\frac{(r+1)^{2}}{r}$

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