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4-2.Quadratic Equations and Inequations
hard
The locus of the point $P=(a, b)$ where $a, b$ are real numbers such that the roots of $x^3+a x^2+b x+a=0$ are in arithmetic progression is
A
an ellipse
B
a circle
C
a parabola whose vertex is on the $Y$-axis
D
a parabola whose vertex is on the $X$-axis
(KVPY-2011)
Solution
(c)
We have, $x^3+a x^2+b x+a=0$
Let $\alpha, \beta, \gamma$ are roots of equation and $\alpha, \beta, \gamma$ are in AP.
Put the values of $\beta$ and $\alpha \gamma$ in Eq. (ii), we get
$\beta(\alpha+\gamma)+\gamma \alpha =b \Rightarrow 2 \beta^2+3=b$
$\Rightarrow 2\left(\frac{a^2}{9}\right)+3 =b \Rightarrow 2 a^2+27=9 b$
$\Rightarrow 2 a^2 =9 b-27$
$\therefore$ Locus of $P(a, b)$ is $2 x^2=9 y-27$ which represent the parabola whose vertex on $Y$-axis.
Standard 11
Mathematics
