Answer to Question #116820 in Analytic Geometry for desmond

Question

Given that z = 1 + i√2, express in the form a + ib each of the complex numbers
p = z + 1/z, q = z − 1/z. In an Argand diagram, P and Q are the points which
represent p and q respectively, O is the orgin, M is the midpoint of P Q and G is the
point on OM such that OG = 2
3
OM. Prove that angle P GQ is a right angle

Answer 1

$$p=z+\frac1z=1+i\sqrt2+\frac1{1+i\sqrt2}$$

Multiply the numerator and denominator of the fraction by $1-i\sqrt2$

$$p=1+i\sqrt2+\frac{1-i\sqrt2}{3}=\frac43+\frac{2\sqrt2}{3}i$$

Similarly, we do with $q$

$$q=z-\frac1z=1+i\sqrt2-\frac1{1+i\sqrt2}=1+i\sqrt2-\frac{1-i\sqrt2}{3}=\frac23+\frac{4\sqrt2}{3}i$$

In this diagram, the point $P$ has coordinates ($\frac43$, $\frac{2\sqrt2}{3}$) , and $Q$ ($\frac23$, $\frac{4\sqrt2}{3}$) . Find the coordinates $M$

$$M_x=\frac{P_x+Q_x}2=1$$ $$M_y=\frac{P_y+Q_y}2=\sqrt2$$

Find the coordinates $G$

$$G_x=M_x\cdot\frac23=\frac23$$

$$G_y=M_y\cdot\frac23=\frac{2\sqrt2}3$$

We define two vectors $\overrightarrow{GP}(GP_x,GP_y)$ and $\overrightarrow{GQ}(GQ_x,GQ_y)$ , if their scalar product is 0, then the angle between them is $90\degree$

$$GP_x=P_x-G_x=\frac23$$

$$GP_y=P_y-G_y=0$$

$$GQ_x=Q_x-G_x=0$$

$$GQ_x=Q_y-G_y=\frac{2 \sqrt2}3$$

$\overrightarrow{GP}\cdot\overrightarrow{GQ}=GP_x\cdot GQ_x+GP_y\cdot GQ_y=0$ ------> $\angle PGQ$ is a right angle