The locus of the mid-point of the focal distance of a variable point moving on the parabola, y2  =  4ax   is a parabola whos

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The given equation represents a parabola with focus at (a,0) and directrix x = -a.

Let P(x1, y1) be a point on the parabola y^2 = 4ax.

The focal distance of P is the distance from P to the focus, which is sqrt((x1-a)^2 + y1^2), and the distance from P to the directrix is |x1 + a|.

The midpoint of the focal distance is the point that is halfway between the focus and the directrix.

Let's denote this midpoint as M. The x-coordinate of M is (x1 + a)/2 and the y-coordinate is y1/2.

The locus of M as P moves along the parabola is the set of all possible positions of M. To find this locus, we eliminate x1 and y1 between the equations of P and M.

From P's equation, we have y1 = 2sqrt(ax1). Substituting this into M's y-coordinate gives us y = sqrt(ax).

Substituting y1 = 2sqrt(ax1) into M's x-coordinate gives us x = (x1 + a)/2. Squaring both sides and simplifying gives us x1 = 4ax - a^2.

Substituting y1 = 2sqrt(ax1) into this equation gives us y = sqrt(4ax - a^2).

So, the locus of M is the parabola y = sqrt(4ax - a^2).

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