Question

If the roots of $a_1{x}^2+b_{1}x+c_1=0$ are ${\alpha }_1,{\beta }_1$ and those of $a_2{x}^2+b_{2}x+c_2=0$ are ${\alpha }_2,{\beta }_2$ such that ${\alpha }_1{\beta }_1=1={\alpha }_2{\beta }_2$ then

• A. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
• B. $\frac{a_1}{c_2}=\frac{b_1}{b_2}=\frac{c_1}{a_2}$
• C. $a_1{a}_2=b_1{b}_2=c_1{c}_2$
• D. None of the above

Answer 1

The correct option is A $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Let $\alpha ,\beta$ are common roots of the quadratic equations $a_1{x}^2+b_{1}x+c_2=0$

Given, $({\alpha }_1{\beta }_1=1={\alpha }_2{\beta }_2)$

Then,
$\alpha +\beta =\frac{{-b}_1}{a_1},\alpha \beta =\frac{c_1}{a_1}$

$\alpha +\beta =\frac{{-b}_2}{a_2},\alpha \beta =\frac{c_2}{a_2}$

$\Rightarrow \frac{{-b}_1}{a_1}=\frac{b_2}{a_2}and\frac{c_1}{a_1}=\frac{c_2}{a_2}$

$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}and\frac{a_1}{a_2}=\frac{c_1}{c_2}$

$\Rightarrow \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$