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Definition of Functions

If |z-3|= m...

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 either $| z - 3 | = | z - 1 |$
$(x - 1)^{2} + y^{2} = (x - 3)^{2} + y^{2}$
$x - 1 = \pm (x - 3)$
$x - 1 = - x + 3$
$x = 2$ ...(i)
Hence $R e (z) = 2$ .
OR
$| z - 3 | = | z - 5 |$
$(x - 3)^{2} + y^{2} = (x - 5)^{2} + y^{2}$
$(x - 3) = \pm (x - 5)$
$x - 3 = - x + 5$
$2 x = 8$
$x = 4$ .
$R e (z) = 4$ .

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