Question

Greatest ratio of roots of $4x^2-2(a^2+b^2)x+a^2{b}^2=0$ if $a=2b$

Answer 1

$4x^2-2(a^2+b^2)x+a^2{b}^2=0$

$\alpha +\beta =\frac{a^2+b^2}{4}$

$\Rightarrow {\alpha }^2+{\beta }^2=\frac{a^4+b^4+2a^2{b}^2}{16}-\frac{a^2{b}^2}{8}=\frac{a^4+b^4}{16}$

$\alpha \beta =\frac{a^2{b}^2}{4}$

$a=2b$

$\therefore \alpha +\beta =\frac{5b^2}{4}$

$\therefore \alpha \beta =\frac{4b^4}{4}=b^4$

$\therefore {\alpha }^2+{\beta }^2=\frac{17b^4}{16}$

$\therefore \frac{\alpha }{\beta }=4$