The locus of the point P=(a, b) where a, b ar | vedclass

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(C) Let the roots of the equation $x^3+a x^2+b x+a=0$ be $\alpha - d, \alpha, \alpha + d$.From the properties of roots,the sum of roots is $(\alpha -

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a parabola whose vertex is on the $Y$-axis

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(C) Let the roots of the equation $x^3+a x^2+b x+a=0$ be $\alpha - d, \alpha, \alpha + d$.From the properties of roots,the sum of roots is $(\alpha - d) + \alpha + (\alpha + d) = 3\alpha = -a$,so $\alpha = -a/3$.Since $\alpha$ is a root,it must satisfy the equation: $(-a/3)^3 + a(-a/3)^2 + b(-a/3) + a = 0$.$-a^3/27 + a^3/9 - ab/3 + a = 0$.Multiplying by $27$,we get $-a^3 + 3a^3 - 9ab + 27a = 0$,which simplifies to $2a^3 - 9ab + 27a = 0$.Since $a 
eq 0$ (if $a=0$,the equation becomes $x^3+bx=0$,roots are $0, \pm \sqrt{-b}$,which are in $AP$ only if $b=0$,but $a=0, b=0$ is a point),we divide by $a$: $2a^2 - 9b + 27 = 0$.Thus,$2a^2 = 9b - 27$,or $b = \frac{2}{9}a^2 + 3$.Replacing $(a, b)$ with $(x, y)$,the locus is $y = \frac{2}{9}x^2 + 3$,which is a parabola with its vertex on the $Y$-axis at $(0, 3)$.

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

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