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Find the locus of the foot of the perpendicular from the origin to a variable straight line which always passess through a fixed point (a,b)

Find the locus of the foot of the perpendicular from the origin to a variable straight line which always passess through a fixed point (a,b)

Grade:12

1 Answers

Avigyan Samanta
30 Points
6 years ago
Let the line passing through (a,b) be,
y= mx-am+b, ….....................................(i)
Now a line perpendicular to this passing through origin is,
y=-x/m,.......................................................(ii)
let the feet of the perpendicular be (h,k)
From (i)
k= mh-am+b,.....................................(iii)
From (ii)
m= -h/k,
Replacing this in (iii)
k= (-h/k)*h + a(h/k) +b,
hence, k2=-h2+ah+bk,
So h2+k2= ah+bk,
 
Hence the locus is x2+y2=ax+by

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