If three distinct numbers a,b,c are in G.P., and the equations ax^2+2bx+c=0 and dx^2+2ex+f…

Question

If three distinct numbers $a,b,c$ are in G.P., and the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root, then which one of the following statements is correct?

Accepted Answer

Since $b^2=ac$: $ax^2+2bx+c=0 \Rightarrow (\sqrt ax+\sqrt c)^2=0 \Rightarrow x=-\sqrt{c/a}$ (repeated root).

Substituting into $dx^2+2ex+f=0$:
$$d\cdot\frac ca - 2e\sqrt{\frac ca}+f=0 \implies \frac da-\frac{2e}{\sqrt{ac}}+\frac fc=0 \implies \frac da+\frac fc = \frac{2e}{b}.$$

So $\dfrac da,\dfrac eb,\dfrac fc$ are in A.P.

Solutions · 1