Question

If the ratio of the roots of $x^2+ax+b=0$ and that of $x^2+cx+d=0$ are equal, then

• A. $a{d}^2=b^{2}c$
• B. $b{d}^2=a^{2}c$
• C. $b^{2}d=a{c}^2$
• D. $a^{2}d=b{c}^2$

Answer 1

The correct option is D $a^{2}d=b{c}^2$

Let ${\alpha }_1,{\beta }_1$ be the roots of $x^2+ax+b=0$ and ${\alpha }_2,{\beta }_2$ be the roots of $x^2+cx+d=0.$
${\alpha }_1+{\beta }_1=-a,$ ${\alpha }_1{\beta }_1=b$
${\alpha }_2+{\beta }_2=-c,$ ${\alpha }_2{\beta }_2=d$

Given that $\frac{{\alpha }_1}{{\beta }_1}=\frac{{\alpha }_2}{{\beta }_2}$
$\Rightarrow \frac{{\alpha }_1+{\beta }_1}{{\alpha }_1-{\beta }_1}=\frac{{\alpha }_2+{\beta }_2}{{\alpha }_2-{\beta }_2}$
$\Rightarrow \frac{({\alpha }_1+{\beta }_1{)}^2}{({\alpha }_1-{\beta }_1{)}^2}=\frac{({\alpha }_2+{\beta }_2{)}^2}{({\alpha }_2-{\beta }_2{)}^2}$
$\Rightarrow \frac{({\alpha }_1+{\beta }_1{)}^2}{({\alpha }_1+{\beta }_1{)}^2-4{\alpha }_1{\beta }_1}=\frac{({\alpha }_2+{\beta }_2{)}^2}{({\alpha }_2+{\beta }_2{)}^2-4{\alpha }_2{\beta }_2}$
$\Rightarrow \frac{a^2}{a^2-4b}=\frac{c^2}{c^2-4d}$
$\Rightarrow a^{2}d=b{c}^2$