Sequence And Series Question 456
Question: The equation $ ( a^{2}+b^{2} )x^{2}-2b( a+c )x+ $ $ ( b^{2}+c^{2} )=0 $ has equal roots. Which one of the following is correct about a, b, and c?
Options:
A) They are in AP
B) They are in GP
C) They are in HP
D) They are neither in AP, nor in GP, nor in HP
Show Answer
Answer:
Correct Answer: B
Solution:
[b] The given equation $ (a^{2}+b^{2})x^{2}-2b(a+c)x+(b^{2}+c^{2})=0 $ has equal roots, so, discriminant = 0 Hence, $ {2b{{(a+c)}^{2}}-4(a^{2}+b^{2})(b^{2}+c^{2})=0 $
$ \Rightarrow ,4b^{2}(a^{2}+c^{2}+2ca)-4(a^{2}b^{2}+a^{2}c^{2}+b^{4}+b^{2}c^{2})=0 $
$ \Rightarrow b^{2}a^{2}+b^{2}c^{2}+2b^{2}ca-a^{2}b^{2}-a^{2}c^{2} $ $ -b^{4}-b^{2}c^{2}=0 $
$ \Rightarrow 2b^{2}ca=b^{4}+a^{2}c^{2} $
$ \Rightarrow b4-2b^{2}ca+a^{2}c^{2}=0 $
$ \Rightarrow {{(b^{2}-ac)}^{2}}=0 $
$ \Rightarrow b^{2}=ac $
$ \Rightarrow $ a, b, c are in GP.
BETA
