Roots of quadratic

Use properties of roots.

D. \({5b}^{2}=36ac\)

For roots we know, \(\alpha+\beta=\frac{-b}{a} \) and \(\alpha\beta=\frac{c}{a} \)

and we know \(\beta =5\alpha\), substituting this in the above relations we get

\(\Rightarrow 6\alpha=\frac{-b}{a} \)    [∵ \(5\alpha+\alpha=6\alpha\)]

\(36{\alpha}^{2}=\frac{{-b}^{2}}{{a}^{2}} ………\text{(i)}\)

\(\Rightarrow\alpha\beta=\frac{c}{a} \)

\({5\alpha}^{2}=\frac{c}{a}\)   [∵ \(\alpha\times5\beta={5\alpha}^{2}\)]

\({\alpha}^{2}=\frac{c}{5a}………\text{(ii)}\)

⇒ combining (i) and (ii) we get,

\(36\times\frac{c}{5a}=\frac{{b}^{2}}{{a}{2}}\)

\(36ac=5{b}^{2}\)