Question

The ratio of the roots of the equation $a_1{x}^2+b_{1}x+c_1=0$ and $a_2{x}^2+b_{2}x+c_2=0$ are equal , then

• A. ${\left(\frac{b_1}{b_2}\right)}^2=\frac{a_2{c}_2}{a_1{c}_1}$
• B. ${\left(\frac{c_1}{c_2}\right)}^2=\frac{b_1{c}_2}{c_1{b}_2}$
• C. ${\left(\frac{a_1}{a_2}\right)}^2={\left(\frac{c_1}{c_2}\right)}^2$
• D. ${\left(\frac{b_1}{b_2}\right)}^2=\frac{a_1{c}_1}{a_2{c}_2}$

Answer 1

The correct option is D ${\left(\frac{b_1}{b_2}\right)}^2=\frac{a_1{c}_1}{a_2{c}_2}$

As $\alpha ,\beta$ are roots of $a_1{x}^2+b_{1}x+c_1=0$
And $\gamma ,\delta$ are roots of $a_2{x}^2+b_{2}x+c_2=0$
$\frac{\alpha }{\beta }=\frac{\gamma }{\delta }\Rightarrow \frac{\alpha }{\gamma }=\frac{\beta }{\delta }$
$\Rightarrow \frac{\alpha +\beta }{\delta +\gamma }=\frac{\beta (\frac{\alpha }{\beta }+1)}{\delta (\frac{\gamma }{\delta }+1)}=\sqrt{{\left(\frac{\beta }{\delta }\right)}^2}=\sqrt{\frac{\alpha \beta }{\gamma \delta }}\Rightarrow \frac{{(\alpha +\beta )}^2}{{(\gamma +\delta )}^2}=\left(\frac{\alpha \beta }{\gamma \delta }\right)$
$\Rightarrow \frac{{(-\frac{b_1}{a_1})}^2}{{(-\frac{b_2}{a_2})}^2}=\frac{\frac{c_1}{a_1}}{\frac{c_2}{a_2}}\therefore \frac{b_1^2}{b_2^2}=\frac{a_1{c}_1}{a_2{c}_2}$