If a,b,c are in G.P then the equations ax^{2} + 2bx + c = 0,dx^{2} + 2ex + f = 0 have a common root if \frac{d}{a},\frac{e}{b},\frac{f}{c} are in: Sequence & Series (Mathematics)

(a) : $a, b, c$ are in G.P, $b=\sqrt{a c} \\ \therefore a x^2+2 \sqrt{a c} x+c=0 (\sqrt{a} x+\sqrt{c})^2 \\=0 \quad \therefore x=-\sqrt{\frac{c}{a}} x=-\sqrt{\frac{c}{a}}\\$ must satisfy the second equation $\\ \therefore d \cdot \frac{c}{a}-2 e \sqrt{\frac{c}{a}}+f=0\\$ dividing by $c, \frac{d}{a}-\frac{2 e}{\sqrt{a c}}+\frac{f}{c}=0 \\ \therefore \frac{2 e}{b}=\frac{d}{a}+\frac{f}{c} \\(\because b =\sqrt{a c}) \therefore \frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.

Explanation

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