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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.
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B
d, e, f are in A.P.
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C
ad,be,cf are in A.P.
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D
ad,be,cf are in H.P.
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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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