Question

Points $z_1$ & $z_2$ are adjacent vertices of a regular octagon. The vertex $z_3$ adjacent to $z_2(z_3\ne z_1)$ can be represented by -

• A. $z_2+\frac{1}{\sqrt{2}}(1\pm i)(z_1+z_2)$
• B. $z_2+\frac{1}{\sqrt{2}}(-1\pm i)(z_1-z_2)$
• C. $z_2+\frac{1}{\sqrt{2}}(-1\pm i)(z_2-z_1)$
• D. none of these

Answer 1

Answer: (B)

$z_2+\frac{1}{\sqrt{2}}(-1\pm i)(z_1-z_2)$

Since

$z_2=z_1{e}^{\pm \frac{i\pi }{4}}$

$z_3=z_2{e}^{\pm \frac{i\pi }{4}}$

Now,

$z_3-z_2=(z_2-z_1)\frac{(1\pm i)}{\sqrt{2}}$

$z_3=z_2+(z_2-z_1)\frac{(1\pm i)}{\sqrt{2}}$

$z_3=z_2+\frac{1}{\sqrt{2}}(-1\pm i)(z_1-z_2)$