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Modulus of a Complex Number

If |z-3|=mi...

Question

If $| z - 3 | = m i n {| z - 1 |, | z - 5 |}$ , then $R e (z)$ equals to

A

2

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B

\frac{5}{2}

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C

\frac{7}{2}

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D

4

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Solution

The correct options are
A 2
D 4
Let $z = x + i y$
Hence
$| z - 3 | = | z - 1 |$
Or
$(x - 3)^{2} + y^{2} = (x - 1)^{2} + y^{2}$
Or
$(x - 3)^{2} - (x - 1)^{2} = 0$
Or
$(2 x - 4) (- 2) = 0$
Hence
$x = 2$ ...(i)
Else
$| z - 3 | = | z - 5 |$
$(x - 3)^{2} = (x - 5)^{2}$
$(x - 3)^{2} - (x - 5)^{2} = 0$
$(2 x - 8) (2) = 0$
$x = 4$ ...(ii)
Hence $R e (z) = 2 o r 4$ .

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