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

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

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

$\Rightarrow \alpha \beta =\frac{c}{a}$

${5\alpha }^2=\frac{c}{a}$   [∵ $\alpha \times 5\beta ={5\alpha }^2$]

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

⇒ combining (i) and (ii) we get,

$36\times \frac{c}{5a}=\frac{b^2}{a2}$

$36ac=5b^2$