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Question

If a,b,c are real numbers and a0. If α is a root of a2x2+bx+c=0 and β is a root of a2x2bxc=0 such that 0<α<β, then the equation a2x2+2bx+2c=0
has a root γ that always satisfies  


Your Answer
A
γ=α+β2
Your Answer
B
γ=αβ2
Your Answer
C
γ=αβ2 
Correct Answer
D
α<γ<β

Solution

The correct option is B α<γ<β
α is a root of  a2x2+bx+c=0
a2α2+bα+c=0bα+c=a2α2...............(1)
Similarly,
bβ+c=a2β2...............(2)
Let f(x)=a2x2+2bx+2c
Then, 
f(α)=a2α2+2bα+2c=a2α2<0                                          ......from(1)f(β)=a2β2+2bβ+2c=3a2β2>0                                           ......from(2)
Thus, f(x) is a polynomial function such that f(α)<0 & f(β)>0
Therefore, there exists γ satifying α<γ<β such that f(γ)=0  
 

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