SOLUTION: If z-2i/2z-1 is purely imaginary prove that the locus of z in the Argands diagram is a circle .find centre and radius

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Question 1099803: If z-2i/2z-1 is purely imaginary prove that the locus of z in the Argands diagram is a circle .find centre and radius
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

This is a curious type of problem that I have not seen before....

Definitely try to get a second opinion....

Let z+=+a%2Bbi

Then %28z-2i%29%2F%282z-1%29+=+%28a%2B%28b-2%29i%29%2F%28%282a-1%29%2B%282b%29i%29

Rationalize the denominator by multiplying by its conjugate:


The condition for that number to be purely imaginary is the the real part must be 0:
2a%5E2-a%2B2b%5E2-4b+=+0

With the a^2 and b^2 terms with the same coefficient, that is indeed the equation of a circle.

To find the center and radius of the circle, complete the squares in a and b:
2%28a%5E2-%281%2F2%29a%2B1%2F16%29+%2B+2%28b%5E2-2b%2B1%29+=+0%2B2%2F16%2B2+=+17%2F8
%28a-%281%2F4%29%29%5E2+%2B+%28b-1%29%5E2+=+17%2F16

The center of the circle is at ((1/4),1); the radius is sqrt(17)/4.