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The locus of the vertices of the family ...

The locus of the vertices of the family of parabolas `y =[a^3x^2]/3 + [a^2x]/2 -2a` is:

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Here, `y = (a^3x^2)/3+(a^2x)/2-2a->(1)`
`=>y +2a = (a^3x^2)/3+(a^2x)/2`
Differentiating both sides w.r.t. `x`,
`=>dy/dx = (2a^3x)/3 +a^2/2`
Let `(alpha,beta)` are the vertices of family of parabola.
Then, `dy/dx|_(alpha,beta) = 0 `
`=> (2a^3alpha)/3 +a^2/2 = 0`
`=>alpha = (-a^2/2)/((2a^3)/3) = -3/(4a)`
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