Q48 / 90 · KVPY 2012 (SB/SX) · Mathematics

Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real numbers, and each of the equations $\mathrm{x}^{2}+2 \mathrm{ax}+\mathrm{b}^{2}=0$ and $\mathrm{x}^{2}+2 \mathrm{bx}+\mathrm{c}^{2}=0$ has two distinct real roots. Then the equation $\mathrm{x}^{2}+2 \mathrm{cx}+\mathrm{a}^{2}=0$ has-

  • A
    Two distinct positive real roots
  • B
    Two equal roots
  • C
    One positive and one negative root
  • Correct answer
    D
    No real roots

Solution

$\mathrm{x}^{2}+2 \mathrm{ax}+\mathrm{b}^{2}=0 \quad \mathrm{x}^{2}+2 \mathrm{bx}+\mathrm{c}^{2}=0$ $\begin{array}{ll}D_{1}>0 & D_{2}>0 \\ 4 a^{2}+b^{2}>0 & 4 b^{2}-4 c^{2}>0 \\ a^{2}>b^{2} \ldots \ldots \ldots .(1) & b^{2}>c^{2} \ldots \ldots \ldots . .(2)\end{array}$ From $(1)$ and $(2)$ $a^{2}>b^{2}>c^{2} \Rightarrow a^{2}>c^{2} \Rightarrow c^{2}-a^{2} < 0$ $x^{2}+2 c x+a^{2}=0$ $D=4 c^{2}-4 a^{2} < 0 \quad$ No real roots

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

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