Help with this question of complex

Dear Student

Given ,|z-4||z-8| =1  and |z-12||z-8i| =53

Putting z = x + iy, we get


|z-4||z-8| = |x+iy-4||x+iy-8| =1|x+iy-4|=|x+iy-8|(x-4)2+y2 =(x-8)2 +y2x2-8x+16 =x2-16x+648x=48x =6

|z-12||z-8i| = 53|x+iy-12||x+iy-8i|=535(|x+iy-8i|) =3|x+iy-12|25(x2+(y-8)2) =9(x-12)2+y2

Putting x = 6, we have

Put x=69(6-12)2+y2=25(62+y-82)9(36+y2) =25(36+y2-6y+64)324+9y2 =2500+25y2-400y16y2-400y+2176 =0y2 -25y+136 =0y2-8y-17y+136 =0y(y-8)-17(y-8)(y-8)(y-17) =0y=8 or y=17

When x = 6 and y = 8, we have

z = 6 + 8i

When x = 6 and y = 17, we have

z = 6 + 17i

Thus, z 1 = 6 + 8i and z 2 = 6 + 17i are two complex numbers satisfying the given equations.


Regards

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