If |z – 5i| = |z + 5i|, then find the locus of z.

1.	If \|z – 5i\| = \|z + 5i\|, then find the locus of z.
Answer» z = a + ib \|a+ib-5i\| = \|a+ib+5i\| \|a+ib-5i\|² = \|a+ib+5i\|² \|a +i(b-5)\|² = \|a + i(b+5)\|² a²+(b-5)² = a²+(b+5)² a²+b²+25-10b = a²+b²+25+10b 20b = 0 b = 0 b is a imaginary part of z Im(z) = \(\frac{z-\bar z}{2}\) = y = 0 Im (z) = 0 So, the locus point is real axis

Answer»

z = a + ib

|a+ib-5i| = |a+ib+5i|

|a+ib-5i|² = |a+ib+5i|²

|a +i(b-5)|² = |a + i(b+5)|²

a²+(b-5)² = a²+(b+5)²

a²+b²+25-10b = a²+b²+25+10b

20b = 0

b = 0

b is a imaginary part of z

Im(z) = \(\frac{z-\bar z}{2}\)

= y

= 0

Im (z) = 0

So, the locus point is real axis

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