First slide

Theory of equations

Question

$If α, β are the non-zero roots of a x^{2} + b x + c = 0 and α^{2}, β^{2} are the roots of a^{2} x^{2} + b^{2} x + c^{2} = 0 then a, b, c are in$

Moderate

A

G.P
B

H.P
C

A.P
D

None of these

Solution

$α, β are roots of a x^{2} + b x + c = 0$
$\begin{array}{l} \begin{array}{l} \Rightarrow α + β = \frac{- b}{a} - - - - - - (1) \\ α β = \frac{c}{a} - - - - (2) \end{array} \\ α^{2}, β^{2} are roots of a^{2} x^{2} + b^{2} x + c^{2} = 0 \end{array}$
$\begin{array}{l} \Rightarrow α^{2} + β^{2} = - \frac{b^{2}}{a^{2}} - - - - - (3) \\ α^{2} \cdot β^{2} = \frac{c^{2}}{a^{2}} - - - - - (4) \end{array}$
$\begin{array}{l} Now from (3) {(α + β)}^{2} - 2 α β = - \frac{b^{2}}{a^{2}} \\ \Rightarrow {(\frac{- b}{a})}^{2} - 2 \frac{c}{a} = \frac{- b^{2}}{a^{2}} \\ \Rightarrow 2 \frac{b^{2}}{a^{2}} = \frac{2 c}{a} \\ \Rightarrow b^{2} = a c \Rightarrow a, b, c are in G.P. \end{array}$

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