Show that the locus of the mid-points of all chords passing through the vertices of the parabola `y^(2)`=4ax is the parabola `y^(2)=2ax`.

Explanation

To show that the locus of the midpoints of all chords passing through the vertices of the parabola \( y^2 = 4ax \) is the parabola \( y^2 = 2ax \), we can follow these steps:

### Step 1: Identify the parabola and its vertex
The given parabola is \( y^2 = 4ax \). The vertex of this parabola is at the origin \( (0, 0) \).

### Step 2: Consider a point on the parabola
Let \( Q(x_1, y_1) \) be a point on the parabola. Since \( Q \) lies on the parabola, it satisfies the equation:
\[
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