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Question

Find the equation to the locus of a point which moves so that its distance from the point (a,0) is always four times its distance from the axis of y.

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Solution

Let the point be P(h,k)
Given point be A(a,0)
PA=(ha)2+(k0)2
Distance of P from y axis be PB
Equation of y axis is x=0
PB=1(h)+0(k)+012+02=h
Given PA=4PB
(ha)2+(k0)2=4h(ha)2+(k)2=16h2h2+a22ah+k2=16h215h2k2a2+2ah=0
Replacing h by x and k by y
15x2y2a2+2ax=015x2y2+2ax=a2
is the required equation of locus

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