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⋅ (x−12)2+9⋅ y2 = 25x2+25⋅ (y−8)2 (x−4)2+y2 = (x−8)2+y2  16⋅ x2 + 216⋅ x + 16 ⋅  y2 − 400 ⋅ y + 304 = 0 x=6  16⋅ x2+216⋅ x+16⋅ y2−400⋅ y+304=0  16⋅ 62+216⋅ 6+16⋅ y2−400⋅ y+304=0 16y2−400y+2176=0 16=24 400=24⋅52 2176=27⋅17 GCD(16,400,2176)=24=16  y2−25y+136=0  a=1;b=−25;c=136 D=b2−4ac=252−4⋅1⋅136=81 D>0  y1,2​=2a−b±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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