Q53 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

If $a, b, c \in R$ are such that $4 a+2 b+c>0$ and $a x^2+b x+c=0$ has no real roots, then the value of $(c+a)(c+b)$ is

  • Correct answer
    A
    greater than ab
  • B
    less than bc
  • C
    greater than ca
  • D
    less than ab + bc + ca

Solution

Since, the equation $a x^2+b x+c=0$ have no real roots and $4 a+2 b+c>0$ $$ \begin{aligned} \therefore \quad a+b+c & >0 \\ & \left\{\because a x^2+b x+c>0, \forall x \in R\right\} \end{aligned} $$ So, $$ \begin{aligned} \text { So, } & & c+a>-b \text { and } c+b & >-a \\ \Rightarrow & & & (c+a)(c+b)>a b \end{aligned} $$

Hence, the correct answer is option: 'A'.

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