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Question: If z<sub>1</sub> = 8 + 4i, z<sub>2</sub> = 6 + 4i, and arg\(\left( \frac{z–z_{1}}{z–z_{2}} \right)\)...

If z1 = 8 + 4i, z2 = 6 + 4i, and arg(z–z1z–z2)\left( \frac{z–z_{1}}{z–z_{2}} \right)= π4\frac{\pi}{4}, then z satisfies –

A

| z – 7 – 4i | = 1

B

| z – 7 – 5i | = 2\sqrt{2}

C

| z – 4i | = 8

D

| z – 7i| = 18\sqrt{18}

Answer

| z – 7 – 5i | = 2\sqrt{2}

Explanation

Solution

Sol. arg (z – z1) – arg (z – z2) = p/4

arg (x – 8 + i (y – 4)) – arg ((x – 6) + i (y – 4)) = p/4

tan–1(y–4x–8)\left( \frac{y–4}{x–8} \right)– tan–1(y–4x–6)\left( \frac{y–4}{x–6} \right)= π4\frac{\pi}{4}

̃ 2(y–4)x2+y2–14x–8y+64\frac{2(y–4)}{x^{2} + y^{2}–14x–8y + 64}= 1

̃ x2 + y2 – 14x – 10 y + 72 = 0

̃ (x – 7)2 + (y – 5)2 = (2)2(\sqrt{2})^{2} ̃ | z – (7 + 5 i) | = 2\sqrt{2}