Sequence And Series Question 187
Question: If the roots of $ a,(b-c)x^{2}+b,(c-a)x+c,(a-b)=0 $ be equal, then $ a,\ b,\ c $ are in
[RPET 1997]
Options:
A) A.P.
B) G.P.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Since the roots of the quadratic equation in $ x $ are equal, we have $ (B^{2}-4AC=0) $
$ \Rightarrow $ $ b^{2}{{(c-a)}^{2}}-4ac(b-c)(a-b)=0 $
$ \Rightarrow $ $ b^{2}(c^{2}-2ca+a^{2})-4ac(ba-b^{2}-ca+bc)=0 $
$ \Rightarrow $ $ b^{2}(c^{2}+2ca+a^{2})-4ac{ b(a+c)-ac }=0 $
$ \Rightarrow $ $ b^{2}{{(a+c)}^{2}}-4ac{b(a+c)-ac}=0 $ which can be seen to be true, if $ b=\frac{2ac}{a+c} $ or $ b(a+c)=2ac $ $ i.e. $ , if $ a,\ b,\ c $ are in H.P.
BETA
