Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2ay=\frac{2a^3}{6}(x^2+\frac{3}{2a}x-\frac{12a}{2a^3})y=\frac{2a^3}{6}(x^2+2.\frac{3}{4a}x+\frac{9}{16a^2}-\frac{9}{16a^2}-\frac{12a}{2a^3})y=\frac{2a^3}{6}({(x+\frac{3}{4a})}^2-\frac{1059}{16a^3})(y+\frac{1059}{48})=\frac{2a^3}{6}{(x+\frac{3}{4a})}^{2}x=\frac{-1059}{48}y=\frac{-3}{49}xy=\frac{1059}{48}\times \frac{3}{49}=\frac{105}{64}$