First slide

Theory of equations

Question

If the ratio of the roots of the equation $x^{2} + b x + c = 0$ is the same as that of the ratio of the roots of $x^{2} + q x + r = 0$ , then

Moderate

A

$b r^{2} = q c^{2}$
B

$c q^{2} = r b^{2}$
C

$q^{2} c^{2} = b^{2} r^{2}$
D

$b q = r c$

Solution

Let $α, β$ be the roots of the equation $x^{2} + b x + c = 0$ and $λ$ , $δ$ be the roots of the equation $x^{2} + q x + r = 0$ .
We are given
$\frac{α}{β} = \frac{γ}{δ} \Rightarrow \frac{α - β}{α + β} = \frac{γ - δ}{γ + δ}$
$\begin{array}{l} \Rightarrow {(\frac{α - β}{α + β})}^{2} = {(\frac{γ - δ}{γ + δ})}^{2} \\ \Rightarrow \frac{(α + β)^{2} - 4 α β}{(α + β)^{2}} = \frac{(γ + δ)^{2} - 4 γ δ}{(γ + δ)^{2}} \\ \Rightarrow \frac{α β}{(α + β)^{2}} = \frac{γ δ}{(γ + δ)^{2}} \\ \Rightarrow \frac{c}{(- b)^{2}} = \frac{r}{(- q)^{2}} \Rightarrow c q^{2} = r b^{2} \end{array}$

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