Q53 / 90 · KVPY 2011 (SB/SX) · Mathematics

The locus of the point $\mathrm{P}=(\mathrm{a}, \mathrm{b})$ where $\mathrm{a}, \mathrm{b}$ are real numbers such that the roots of $\mathrm{x}^{3}+\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{a}=0$ are in arithmetic progression is-

  • A
    an ellipse
  • B
    a circle
  • Correct answer
    C
    a parabola whose vertex in on the $y$-axis
  • D
    a parabola whose vertex is on the $\mathrm{x}$-axis

Solution

Let roots $\alpha-\mathrm{d}, \alpha, \alpha+\mathrm{d}$ $\operatorname{Sum} 3 \alpha=-\mathrm{a} \Rightarrow \alpha=-\frac{a}{3}$ product $\alpha\left(\alpha^{2}-\mathrm{d}^{2}\right)=-\mathrm{a}$ $\begin{aligned} \text { pair product } b &=\alpha^{2}-\alpha d+\alpha^{2}+\alpha d+\alpha^{2}-d^{2} \quad \alpha^{2}-d^{2}=3 \\ b &=2 \alpha^{2}+3 \\ & b-3=\frac{2}{9} a^{2} \Rightarrow \text { locus } x^{2}=\frac{9}{2}(y-3) \text { parabola } \end{aligned}$

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

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