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.

Answer 1

a)

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

$z=x+iy$

$$4((x+1)^2+y^2)=(x-2)^2+y^2$$ $$4x^2+8x+4+4y^2=x^2-4x+4+y^2$$ $$3x^2+12x+3y^2=0$$ $$x^2+4x+4-4+y^2=0$$ $$(x+2)^2+y^2=4$$

The equation of the circle with center $(-2,0)$ and radius $2.$

b)

$$|{z+i \over z-5-2i}|=1$$ $$|z+i|=|z-5-2i|,z\not=5+2i$$

$z=x+iy$

$$x^2+(y+1)^2=(x-5)^2+(y-2)^2$$ $$x^2+y^2+2y+1=x^2-10x+25+y^2-4y+4$$ $$6y=-10x+28$$

$$y=-{5 \over 3}x+{14 \over 3}$$

c)

$$Im(z+{9\over z})=0$$

$z=x+iy, z\not=0$

$$x+iy+{9\over x+iy}=x+iy+{9\over x^2+y^2}(x-iy)$$

$$y(x^2+y^2)-9y=0$$

$$y=0 \ or \ x^2+y^2=9$$

Real axis or the circle with center $(0,0)$ and radius $3$ with the exception of point $(0,0).$