First slide

Geometry of complex numbers

Question

$If ' z'lies on the circle | z - 3 i | = 3 \sqrt{2}, then the value of \arg (\frac{z - 3}{z + 3}) is equal to$

Moderate

A

$\frac{π}{3}$
B

$\frac{π}{4}$
C

$\frac{π}{6}$
D

$\frac{π}{2}$

Solution

$\begin{array}{l} Given ' z'lies on the circle | z - 3 i | = 3 \sqrt{2} \\ C A = C B = 3 \sqrt{2}, O C = 3 \\ \Rightarrow O A = O B = 3 \\ \Rightarrow A = 3 + 0 i, B = - 3 + 0 i \\ Clearly, ∠ B C A = \frac{π}{2} \\ \Rightarrow ∠ B P A = \frac{π}{4} \\ \Rightarrow \arg (\frac{z - 3}{z + 3}) = \frac{π}{4} \end{array}$

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