Question

The ratio of the roots of the $a_1{x}^2+b_{1}x+c_1=0$ be equal to the ratio of roots of $a_2{x}^2+b_{2}x+c_2=0$ then $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

• A. True
• B. False

Answer 1

The correct option is A True

Let $\alpha ,\beta$ be the common roots of the quadratic equations $a_1{x}^2+b_{1}x+c_1=0$ and $a_2{x}^2+b_{2}x+c_2=0$.

Then,

$\alpha +\beta =\frac{-b_1}{a_1},\alpha \beta =\frac{c_1}{a_1}$

And $\alpha +\beta =\frac{-b_2}{a_2},\alpha \beta =\frac{c_2}{a_2}$

Therefore,

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

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

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

Hence, this is the answer.