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Ifthe roots of the equation $3 \mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0$ are in the ratio $2: 3$, then which one of the following is correct?
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The correct answer is:
$8 b^{2}=25 a c$
Let roots of equation be $2 \alpha$ and $3 \alpha$
$2 \alpha+3 \alpha=-\frac{2 b}{3 a}$
$\Rightarrow \alpha=\frac{-2 b}{15 a}$...(i)
$2 \alpha \cdot 3 \alpha=\frac{\mathrm{c}}{3 \mathrm{a}} \Rightarrow \alpha^{2}=\frac{\mathrm{c}}{18 \mathrm{a}}$...(ii)
Now, put value of $\alpha$ in equation (ii) $\left(\frac{-2 b}{15 a}\right)^{2}=\frac{c}{18 a} \Rightarrow \frac{4 b^{2}}{225 a^{2}}=\frac{c}{18 a}$
$\Rightarrow 8 b^{2}=25 a c$
$2 \alpha+3 \alpha=-\frac{2 b}{3 a}$
$\Rightarrow \alpha=\frac{-2 b}{15 a}$...(i)
$2 \alpha \cdot 3 \alpha=\frac{\mathrm{c}}{3 \mathrm{a}} \Rightarrow \alpha^{2}=\frac{\mathrm{c}}{18 \mathrm{a}}$...(ii)
Now, put value of $\alpha$ in equation (ii) $\left(\frac{-2 b}{15 a}\right)^{2}=\frac{c}{18 a} \Rightarrow \frac{4 b^{2}}{225 a^{2}}=\frac{c}{18 a}$
$\Rightarrow 8 b^{2}=25 a c$
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