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Question

Find the complex number z satisfying the equations z12z8i=53, z4z8=1


A

6

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B

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C

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D

None of these

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Solution

The correct option is C


We have z12z8i=53, z4z8=1

Let z=x+iy, then

z12z8i=53 3|z-12| = 5|z-8i|

3|(x-12) + iy| = 5|x + (y-8)i|

9(x12)2+9y2 = 25x2+25(y8)2 (i)

z4z8=1 |z-4| = |z-8|

(x-4 + iy| = |x -8+iy|

(x4)2+y2 = y2+(x8)2 x=6

Putting x=6 in (i), we get y2-25y+136=0

y=17, 8

Hence z=6+17i or z=6+8i

Trick: Check it with options.


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