The locus of the vertices of the family of parabolas y=a^3x^23+a^2x2-2a is:

Question

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

Accepted Answer

Complete the square:
$$y=\frac{a^3}{3}\left(x+\frac3{4a}\right)^2-\frac{35a}{16}.$$
Vertex: $x=-\dfrac{3}{4a},\ y=-\dfrac{35a}{16}$.

From these, $a=-\dfrac{3}{4x}$ and $a=-\dfrac{16y}{35}$. Equating:
$$-\frac{3}{4x}=-\frac{16y}{35}\Rightarrow 105=64xy\Rightarrow xy=\frac{105}{64}.$$

Solutions · 1