Let $\alpha,\beta$ be the roots of the equation ${ax}^{2} + 2bx + c = 0$ and $\gamma,\delta$ be the

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Let $\alpha,\beta$ be the roots of the equation ${ax}^{2} + 2bx + c = 0$ and $\gamma,\delta$ be the roots of the equation ${px}^{2} + 2qx + r = 0$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then

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Sol. [Key Concept: Roots of two quadratics all in G.P. ⇒ relate the coefficients via the common ratio.]$$\alpha(1 + d) = \frac{2b}{a}\ \alpha^{2}d = \frac{c}{a}$$$$\alpha d^{2}(1 + d) = - \frac{2q}{p}\ \alpha^{2}d^{5} = \frac{r}{p}$$$${\Rightarrow \ d^{2} = \frac{qa}{bp}\ d^{4} = \frac{ra}{cp} }{\Rightarrow \frac{q^{2}a^{2}}{b^{2}p^{2}} = \frac{ra}{cp} \Rightarrow \frac{q^{2}ac}{b^{2}} = pr \Rightarrow \frac{pr}{q^{2}} = \frac{ac}{b^{2}}}$$

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