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Question

Find the locus of P for which the distance from P to origin is double the distance from P to the point (1,2).

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Solution

Let the point is P(h,k)
Distance of P from origin =(h0)2+(k0)2
=h2+k2

Distance of P from the point (1,2) =(h1)2+(k2)2
=h2+12h+k2+44k
=h22h+k24k+5

Now,
h2+k2=2h22h+k24k+5

Squaring both sides,
h2+k2=4(h22h+k24k+5)

h2+k2=4h2+4k28h16k+20

3h2+3k28h16k+20=0

So, the locus of P is
3x2+3y28x16y+20=0

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