Question

# If a, b, c are rational numbers (a>b>c>0) and the quadratic equation (a+b−2c)x2+(b+c−2a)x+(c+a−2b)=0 has a root in the interval (−1,0), then which of the following statements is (are) CORRECT?a+c<2bBoth roots are rationalax2+2bx+c=0 has both roots negativecx2+2bx+a=0 has both roots negative

Solution

## The correct options are A a+c<2b B Both roots are rational C ax2+2bx+c=0 has both roots negative D cx2+2bx+a=0 has both roots negative(a+b−2c)x2+(b+c−2a)x+(c+a−2b)=0, a+b−2c≠0 The given equation has one root as 1. Let α be the other root. 1⋅α=c+a−2ba+b−2c So, the other root is x=c+a−2ba+b−2c  As c+a−2ba+b−2c<0 and a+b>2c ⇒c+a−2b<0 ⇒a+c<2b For equation ax2+2bx+c=0, f(0)=c>0,−2ba<0  So both roots are negative. Similarly, for cx2+2bx+a=0, both roots are negative.

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