Complex Numbers And Quadratic Equations question 195
Question: If $ a,b,c $ are in G.P., then the equations $ ax^{2}+2bx+c=0 $ and $ dx^{2}+2ex+f=0 $ have a common root if $ \frac{d}{a},\frac{e}{b},\frac{f}{c} $ are in [IIT 1985; Pb. CET 2000; DCE 2000]
Options:
A) A.P.
B) G.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
As given, $ b^{2}=ac $
Þ equation $ ax^{2}+2bx+c=0 $ can be written as $ ax^{2}+2\sqrt{ac}x+c=0 $
Þ $ {{(\sqrt{a}x+\sqrt{c})}^{2}}=0 $
Þ $ x=-\sqrt{\frac{c}{a}} $ (repeated root) This must be the common root by hypothesis. So it must satisfy the equation $ dx^{2}+2ex+f=0 $
Þ $ d\frac{c}{a}-2e\sqrt{\frac{c}{a}}+f=0 $
Þ $ \frac{d}{a}+\frac{f}{c}=\frac{2e}{c}\sqrt{\frac{c}{a}}=\frac{2e}{b} $
Þ $ \frac{d}{a},\frac{e}{b},\frac{f}{c} $ are in A.P.
BETA
