1DIAMETER OF THE CIRCLE
d locus of middle points of chords of d parabola y2 =4ax which r such dat d normals at their xtremities meet on d parabola is
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10 Answers
1itz a straight line.....thats for sure.........[3][3][3].........but i dont think we cant say it the diamter of the circle........
1YA SRY I MISTAKENLY WROTE TAT YA ITS TRUE ITS DIAMETER OF PARABOLA :D
1Check this out>>>
http://chestofbooks.com/architecture/American-House-Carpenter/The-Parabola-Part-2.html
1assume the two pts to be t1 and t2....(parametrically) and the pt at which their normals meet to be t
now we have ,
2/(t+t1) =-t1-----1) (slope of the t1,t line)
and also 2/(t+t2)=-t2----2)
let the midpt be (h,k)
we have h=(a(t12+t22)/2)-----6)
and k=(a(t1+t2))-------5)
from 1 and 2, we have
-2/t1 =t+t1------3)
and
-2/t2=t+t2------4)
now subtracting 4) from 3) we have
2(1/t2 - 1/t1) =t1-t2
or 2/t1t2 =1
t1t2=1
now from 5 and 6 we know that t1t2= [{k/a}2-2(h/a)]/2
so we have the reqd locus is that,
4a2=k2-2ah...
this cud have been done much easily if one knew the property of t1t2=2 from before...