Find all 2

Find all the complex solutions (write the answer in the form x+iy) of the system.
{|z-12|/|z-8i|=5/3 ; |z-4|=|z-8|

Correct answer:

z1 = 6 + 8 i
z2 = 6 + 17 i

Step-by-step explanation:

z = x+i y 3 dist(z,12) = 5 dist(z,8i) dist(z,4) = dist(z,8)  9 (x12)2+9 y2 = 25x2+25 (y8)2 (x4)2+y2 = (x8)2+y2  16 x2 + 216 x + 16   y2  400  y + 304 = 0 x=6  16 x2+216 x+16 y2400 y+304=0  16 62+216 6+16 y2400 y+304=0 16y2400y+2176=0 16=24 400=2452 2176=2717 GCD(16,400,2176)=24=16  y225y+136=0  a=1;b=25;c=136 D=b24ac=25241136=81 D>0  y1,2=2ab±D=225±81 y1,2=225±9 y1,2=12.5±4.5 y1=17 y2=8  z1=x+i y1 z1=6+8 i

Our quadratic equation calculator calculates it.

z2=x+i y2  z2=6+17 i

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