Question

If a, b, c are in G.P. and the equations $$ax^2+2bx+c=0=0$$ and $$dx^2+2ex+f=0$$ have a common root. Then?

A
d, e, f are in G.P.
B
d, e, f are in A.P.
C
D

Solution

The correct option is D $$\dfrac{a}{d}, \dfrac{b}{e}, \dfrac{c}{f}$$ are in H.P.$$b^2=ac$$roots of $$ax^2+2bx+c=0$$ are equal i.e., $$-\dfrac{b}{a}$$$$d\left(-\dfrac{b}{a}\right)^2+2e\left(-\dfrac{b}{a}\right)+f=0$$$$db^2-2bea+fa^2=0$$$$dc-2eb+fa=0$$divide by ac$$\dfrac{dc}{ac}-\dfrac{2eb}{b^2}+\dfrac{fa}{ac}=0$$$$\Rightarrow \dfrac{d}{a}-\dfrac{2eb}{b^2}+\dfrac{fa}{ac}=0$$$$\Rightarrow \dfrac{d}{a}-\dfrac{2e}{b}+\dfrac{f}{c}=0$$$$\dfrac{d}{a}, \dfrac{e}{b}, \dfrac{f}{c}$$ are in A.P.Mathematics

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