If z in C, then what does the equation 2|z+3i|-|z-i|=0 represent?

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1 Answer
Mar 18, 2018

This is the equation of a circle, center (0, -13/3) and radius =8/3

Explanation:

The equation is

2|z+3i|=|z-i|

The modulus of (z-i) is twice the modulus of (z+3i)

Let z=x+iy

Then,

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

2|x+i(y+3)|=|x+i(y-1)|

Then,

2sqrt(x^2+(y+3)^2)=sqrt(x^2+(y-1)^2)

Squaring both sides

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

4(x^2+y^2+6y+9)=(x^2+y^2-2y+1)

4x^2-x^2+4y^2-y^2+24y+2y+36-1

3x^2+3y^2+26y+35=0

=3x^2+3(y^2+26/3y+169/9)=-35+3*169/9

=x^2+(y+13/3)^2=64/9=(8/3)^2

This is the equation of a circle, center (0, -13/3) and radius =8/3

graph{x^2+(y+13/3)^2-64/9=0 [-7.6, 10.18, -7.55, 1.34]}