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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}} \righ...

If z1 = 8 + 4i, z2 = 6 + 4i and arg (z−z1z−z2)=π4\left( \frac{z - z_{1}}{z - z_{2}} \right) = \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. If z−z1z−z2=(x−8)+i(y−4)(x−6)+i(y−4)\frac{z - z_{1}}{z - z_{2}} = \frac{(x - 8) + i(y - 4)}{(x - 6) + i(y - 4)}

Arg (z−z1z−z2)=π4\left( \frac{z - z_{1}}{z - z_{2}} \right) = \frac{\pi}{4}

Q Arg (z – z1) – Arg (z – z2) = π4\frac{\pi}{4}

tan–1(y−4x−8)\left( \frac{y - 4}{x - 8} \right) – tan–1 (y−4x−6)=π4\left( \frac{y - 4}{x - 6} \right) = \frac{\pi}{4} z(y−4)x2+y2−14x−8y+64\frac{z(y - 4)}{x^{2} + y^{2} - 14x - 8y + 64} = 1

x2 + y2 – 14x – 10y + 72 = 0

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