The locus of the vertices of the family of parabolas $y=\frac{a^3 x^2}{3}+\frac{a^2 x}{2}-2 a$ is

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Aicharya · Accepted Answer

Parabola: $y=\frac{a^3 x^2}{3}+\frac{a^2 x}{2}-2 a$ Vertex: $(\alpha, \beta)$ $ \begin{aligned} & \alpha=\frac{-a^2 / 2}{2 a^3 / 3}=-\frac{3}{4 a}, \beta=\frac{-\left(\frac{a^4}{4}+4 \cdot \frac{a^3}{3} \cdot 2 aight)}{4 \frac{a^3}{3}}=-\frac{-\left(\frac{1}{4}+\frac{8}{3}ight) a^4}{\frac{4}{3} a^3} \ & =-\frac{35}{12} \frac{a}{4} 	imes 3=-\frac{35}{16} a \ & \alpha \beta=-\frac{3}{4 a}\left(-\frac{35}{16}ight) a=\frac{105}{64} . \end{aligned} $

The locus of the vertices of the family of parabolas $y=\frac{a^3 x^2}{3}+\frac{a^2 x}{2}-2 a$ is

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