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How could I find the locus of M as P moves of the parabola. P is.(2at, at^2) . M is the midpoint of the x and y intercepts of the normal through P.

So far I was able to find the quation of the normal and thus was able to fund the midpoint. However I don't really i don't really understand how to find the locus. By the way I have a lot of trouble with my locus so if possible can you please give me up for links or instructions on how to find a locus, in general.

All help is appreciated!

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equation is $x^2=4ay$ for $(2at,at^2)$

slope of tangent $$2x=4ay'$$ $$y'=t$$ slope of normal $$-1/t$$ eqn of normal $$y-at^2=-1/t(x-2at)$$ $$ty+x=2at+at^3$$ when $x=0$ $$y=2a+at^2$$ when $y=0$ $$x=2at+at^3$$ M is at $(h,k)\equiv(\frac12(2at+at^3),\frac12(2a+at^2))$

now $h/k=t$ substitute: $$2k=2a+ah^2/k^2$$ $$2k^3=2ak^2+ah^2$$

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