Answer to Question #118256 in Complex Analysis for jacob

Question

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|

Answer 1

$2|z+1|=|z-2|$

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

$2^2|(x+1)+iy|^2=|(x-2)+iy|^2$

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

$4x^2+8x+4+4y^2=x^2-4x+4+y^2$

$3x^2+12x+3y^2=0$

$x^2+4x+y^2=0$

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

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

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