Q53 / 160 · AP EAMCET 2024 (21 May Shift 2) · Mathematics

The condition that the roots of $x^3-b x^2+c x-d=0$ are in arithmetic progression is

  • Correct answer
    A
    $9 c b=2 b^3+27 d$
  • B
    $9 c b=2 d^3+27 b$
  • C
    $9 c d=2 d^3+27 b$
  • D
    $9 c d=2 b^3+27 d$

Solution

Let roots in A.P. be as $\alpha-r, \alpha, \alpha+r$ Sum of roots $=b$ $\Rightarrow \alpha-r+\alpha+\alpha+r=b \Rightarrow \alpha=\frac{b}{3}$ $\alpha$ is root of $x^3-b x^2+c x-d=0$ $\Rightarrow \frac{b^3}{27}-\frac{b^3}{9}+\frac{b c}{3}-d=0 \Rightarrow 9 c b=2 b^3+27 d$

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

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