z = 4i
sub in
|4i - 3| = |4i + i| = |4i - 1 - 2i| = |4i - 3| = |5i| = |2i - 1| so
z = i and 5
+118703 |z - 3| = |z + i| = |z - 1 - 2i|.
let z= a+bi
|a+bi−3|=|a+bi+i|=|a+bi−1−2i||(a−3)+bi|=|a+(b+1)i|=|(a−1)+(b−2)i|(a−3)2+b2=a2+(b+1)2=(a−1)2+(b−2)2a2−6a+9+b2=a2+b2+2b+1=a2−2a+1+b2−4b+4−6a+9=2b+1=−2a+1−4b+4−6a+8=2b=−2a−4b+4−3a+4=b=−a−2b+2 b=−3a+4so−3a+4=−a−2(−3a+4)+2−3a+4=−a+6a−8+2−3a+4=5a−6−8a=−10a=1.25b=−3a+4=−3.75+4=0.25z=5+i4
Latex:
|a+bi - 3| = |a+bi + i| = |a+bi - 1 - 2i| \\
|(a-3)+bi| = |a+(b+1)i| = | (a -1)+(b - 2)i |\\
(a-3)^2+b^2 = a^2+(b+1)^2 = (a-1)^2+(b-2)^2\\
a^2-6a+9+b^2=a^2+b^2+2b+1=a^2-2a+1+b^2-4b+4\\
-6a+9=2b+1=-2a+1-4b+4\\
-6a+8=2b=-2a-4b+4\\
-3a+4=b=-a-2b+2\\~\\
b=-3a+4\\so\\
-3a+4=-a-2(-3a+4)+2\\
-3a+4=-a+6a-8+2\\
-3a+4=5a-6\\
-8a=-10\\
a=1.25\\
b=-3a+4=-3.75+4=0.25\\
z=\frac{5+i}{4}
