Question

The values of $x,y \in R$ and $(x + iy)(3 + 2i) = 1 + i$ for which the numbers $(x,y)$ and $\left( 1,\frac{1}{5} \right)$ are conjugate complex can be.

• A. $\left( \frac{1}{13},\frac{1}{13} \right)$or $\left( \frac{5}{13},\frac{1}{13} \right)$
• B. $\left( \frac{1}{5},\frac{1}{5} \right)$or $\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib$
• C. $a = 2,b = - 1$or $a = 1,b = 0$
• D. None of these

Answer 1

Answer: (A)

$\left( \frac{1}{13},\frac{1}{13} \right)$or $\left( \frac{5}{13},\frac{1}{13} \right)$

Answer 2

According to condition,

$\frac{- 7 - 26i}{25}$

⇒ $\because$and $a = \overline{a}$ ⇒ $(z + a)(\overline{z} + a) = (z + a)(\overline{z} + \overline{a}) = (z + a)(\overline{z + a})$

⇒ $= |z + a|^{2}$or $\frac{z - i}{z + i} = \frac{x + i(y - 1)}{x + i(y + 1)}.\frac{x - i(y + 1)}{x - i(y + 1)}$