# Complex numbers problems......

• Jan 24th 2010, 11:36 AM
asnali
Complex numbers problems......
a)Show that the points representing the complex numbers7 + 9i, -3 + 7i and 3 +3i form a right angled triangle on the Argand diagram.

b) if w = a + ib and z = x + iy, where a,b,x and y are real numbers, such that w=z / 1 +iz, show that a =x / x^2 +(y-1)^2 and b = -{x^2+y^2 -y / x^2 + (y-1)^2}
• Jan 24th 2010, 11:43 AM
Jhevon
Quote:

Originally Posted by asnali
a)Show that the points representing the complex numbers7 + 9i, -3 + 7i and 3 +3i form a right angled triangle on the Argand diagram.

write each number as 2-D vectors where the real part is the first coordinate and the imaginary part is the second coordinate. draw these on the Argand diagram. show that the magnitude of the vectors obey Pythagoras' theorem.

Quote:

b) if w = a + ib and z = x + iy, where a,b,x and y are real numbers, such that w=z / 1 +iz, show that a =x / x^2 +(y-1)^2 and b = -{x^2+y^2 -y / x^2 + (y-1)^2}
i assume you mean $w = \frac z{1 + iz}$ (use parentheses or learn LaTeX! -- see my signature for the latter)

That is, $w = \frac z{(1 - y) + ix}$

Let $c = (1 - y) + ix$.

Hint: find and simplify $w \cdot \frac {\bar c}{\bar c}$