The locus of the vertex of the family of parabolas `y=(a^3x^2)/3+(a^(2x))/2-2a` is `x y=(105)/(64)` (b) `x y=3/4` `x y=(35)/(16)` (d) `x y=(64)/(105)`

Options

A. xy=105/64
B. xy=3/4
C. xy=35/16
D. xy=64/105

Explanation

To find the locus of the vertex of the given family of parabolas \( y = \frac{a^3 x^2}{3} + \frac{a^2 x}{2} - 2a \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation of the parabola:
\[ y = \frac{a^3 x^2}{3} + \frac{a^2 x}{2} - 2a \]
To make calculations easier, we can multiply through by \( 3a^3 \) to eliminate the fraction:
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