First slide

Algebra of complex numbers

Question

If $x^{2} + y^{2} = 1$ and $x \neq - 1$ then $\frac{1 + y + i x}{1 + y - i x}$ equals

Moderate

A

1
B

$x + i y$
C

2
D

$y + i x$

Solution

TIP
Sometimes, it is helpful to write $x^{2} + y^{2} = z \bar{z}$
where $z = x + i y or y + i x$
Put $z = y + i x,$ then $1 = x^{2} + y^{2} = z \bar{z}$
Now,
$\begin{array}{l} \frac{1 + y + i x}{1 + y - i x} = \frac{z \bar{z} + z}{1 + \bar{z}} = \frac{z (\bar{z} + 1)}{1 + \bar{z}} \\ = z = y + i x \end{array}$

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