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 ? 
 

  • Option 1)

    \frac{d}{a},\frac{e}{b},\frac{f}{c}  are in A.P.

  • Option 2)

    d,e,f are in A.P.

  • Option 3)

    d,e,f are in G.P.

     

  • Option 4)

    \frac{d}{a},\frac{e}{b},\frac{f}{c} are in G.P. 

 

Answers (1)
S solutionqc

Given a,b,c are in G.P.

\Rightarrow b^{2}=ac

Eq:\: ax^{2}+2bx+c=0

D=4b^{2}-4ac=0

roots are = -\frac{b}{a},-\frac{b}{a}

dx^{2}+2ex+f=0

root of this eq = \frac{-b}{a}

\Rightarrow d\left ( \frac{-b}{a} \right )^{2}+2e\left ( \frac{-b}{a} \right )+f=0

\Rightarrow db^{2}-2aeb+a^{2}f=0

\because b^{2}=ac

\Rightarrow d(ac)-2aeb+a^{2}f=0

or    dc-2eb+af=0

divide by ac

\frac{dc}{ac}-\frac{2eb}{ac}+\frac{af}{ac}=0

\Rightarrow \frac{d}{a}-\frac{2eb}{b^{2}}+\frac{f}{c}=0\Rightarrow \frac{d}{a}+\frac{f}{c}=\frac{2e}{b}

\frac{d}{a},\frac{e}{b},\frac{f}{c}\rightarrow AP


Option 1)

\frac{d}{a},\frac{e}{b},\frac{f}{c}  are in A.P.

Option 2)

d,e,f are in A.P.

Option 3)

d,e,f are in G.P.

 

Option 4)

\frac{d}{a},\frac{e}{b},\frac{f}{c} are in G.P. 

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