Question

Find the locus of the vertices of the family of parabola $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$

Answer 1

Family of parabola,

$y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$

$h=\frac{-B}{2A}=\frac{-a^2/2}{2x{a}^3/3}=\frac{-3}{4a}$

$k=\frac{(a^2/2{)}^2-4a^3(-2a)/3}{4\left(a^3/3\right)}$ = $\frac{a^4/4-a^4/3}{4a^3/3}$

$k=\frac{-35a}{16}$

$h\times k=\frac{-3}{4a}\times \frac{-35a}{16}=\frac{105}{64}=xy$