The locus of the $$mid-point$$ of the line segment joining the focus to a moving point on the parabola $$y2=$$ $$4ax$$ is another parabola with directrix

Question

The locus of the $$mid-point$$ of the line segment joining the focus to a moving point on the parabola $$y2=$$ $$4ax$$ is another parabola with directrix

Infinity Learn · Accepted Answer

The focus of the parabola $$y2$$ $$=$$ $$4ax$$ is S (a$$, $$0), let P (at2$$ , $$2at) be any point on the parabola then coordinates of the $$mid-point$$ of SP  are given by $$x=at2+12$$, $$y=2at+02Eliminating$$ ‘t’ we get the locus of the $$mid-point$$ as $$y2=2ax$$−$$a2$$  or  $$y2=2a(x$$−$$a/2)$$          (1)which$$ is a parabola of the form $$Y2=4AX$$   $$(2)where$$ $$Y=y$$,$$X=x$$−$$a/2$$ and $$A=a/2Equation$$ of the directrix of $$(2) is $$X=$$−ASo equation the directrix of (1) is  x−$$a/2=$$−$$a/2$$⇒$$x=0$$

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ is another parabola with directrix

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The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2= 4ax is another parabola with directrix

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The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ is another parabola with directrix