How do i solve? If |z-3i| =2|z-3|. Find the locus of the point represented by z thank you

Question

How to find the locus of the point z

Thanks

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Answer 1 · 2018-03-05T12:46:18

The locus is a circle.

Let the points #z_A=(3i)# and #z_B=3#

The point #z_M# is the point #z=x+iy#

Then,

#|z-3i|= 2|z-3|#

#|x+iy-3i|=2|x+iy-3|#

#sqrt(x^2+(y-3)^2)=2sqrt((x-3)^2+y^2)#

#x^2+y^2-6y+9=4(x^2-6x+9+y^2)#

#3x^2+3y^2+6y-24x+27=0#

#3(x^2-8x+16)+3(y^2+2y+1)+27-48-3=0#

#3(x-4)^2+3(y+1)^2=24#

#(x-4)^2+(y+1)^2=8#

This is the eqaution of a circle #"Center"=(4,-1)# and radius #=sqrt8#

graph{(y/3+x/3-1)((x-4)^2+(y+1)^2-8)=0 [-1.58, 12.47, -1.347, 5.676]}