Answer to Question #118256 in Complex Analysis for jacob

Question #118256
Find the Cartesian equation of the locus of the point P representing the complex
number z. Sketch the locus of P in each case.
(a) 2|z + 1| = |z − 2|
1
Expert's answer
2020-05-26T18:29:11-0400

2z+1=z22|z+1|=|z-2|

2x+iy+1=x+iy22|x+iy+1|=|x+iy-2|

22(x+1)+iy2=(x2)+iy22^2|(x+1)+iy|^2=|(x-2)+iy|^2

4((x+1)2+y2)=(x2)2+y24((x+1)^2+y^2)=(x-2)^2+y^2

4x2+8x+4+4y2=x24x+4+y24x^2+8x+4+4y^2=x^2-4x+4+y^2

3x2+12x+3y2=03x^2+12x+3y^2=0

x2+4x+y2=0x^2+4x+y^2=0

(x2+4x+4)+y2=4(x^2+4x+4)+y^2=4

(x+2)2+y2=4(x+2)^2+y^2=4


Answer: the locus is a circle with center (2,0)(-2,0) and radius 22 .


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