Question

 If |(z-2)/(z-4)| = 1 then Re(Z) is equal to​

Answer 1

SOLUTION

GIVEN

$\displaystyle \sf{ \bigg| \frac{z - 2}{z - 4} \bigg| = 1 }$

TO DETERMINE

The value of Re(Z)

EVALUATION

Let z = x + iy

$\displaystyle \sf{ \bigg| \frac{z - 2}{z - 4} \bigg| = 1 }$

$\displaystyle \sf{ \implies | z - 2 | = | z - 4 | }$

$\displaystyle \sf{ \implies | (x + iy) - 2 | = | (x + iy) - 4 | }$

$\displaystyle \sf{ \implies | (x - 2) + iy | = | (x - 4)+ iy | }$

$\displaystyle \sf{ \implies \sqrt{ {(x - 2)}^{2} + {y}^{2} } = \sqrt{ {(x - 4)}^{2} + {y}^{2} } }$

$\displaystyle \sf{ \implies {(x - 2)}^{2} + {y}^{2} = {(x - 4)}^{2} + {y}^{2} }$

$\displaystyle \sf{ \implies {(x - 2)}^{2} = {(x - 4)}^{2} }$

$\displaystyle \sf{ \implies {x}^{2} - 4x + 4 = {x}^{2} - 8x + 16 }$

$\displaystyle \sf{ \implies 4x = 12}$

$\displaystyle \sf{ \implies x = 3}$

$\displaystyle \sf{ \implies Re(Z) = 3}$

FINAL ANSWER

Re(Z) = 3

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