A.D_normal

What normal to the curve y=x² forms the shortest chord ??

2 Answers

let y=mx+c is the normal
then we have x²=mx+c
x=[mÂ±âˆš(m²+4c)]/2

also we have dy/dx=2x=-1/m=

-1/m=m-âˆš(m²+4c)
or âˆš(m²+4c)=m+1/m

so the points are x=mÂ±(m+1/m)=2m+1/m or -1/m

D²=(x-x')²+(x²-x'²)²=(2m+2/m)²+(4m²+4)²

now find dD/dm

you know that the normal meets again at -t-2/t

find the distance between points with parametric coordinates t1 and t2

now minimize :)

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