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

$$z=1+i\sqrt{2}$$

$$p=z+{1\over z}=1+i\sqrt{2}+{1\over 1+i\sqrt{2}}=1+i\sqrt{2}+{1-i\sqrt{2}\over 1^2+(\sqrt{2})^2}=$$

$$={3+i3\sqrt{2}+1-i\sqrt{2}\over 3}={4+i2\sqrt{2}\over 3}$$

$$q=z-{1\over z}=1+i\sqrt{2}-{1\over 1+i\sqrt{2}}=1+i\sqrt{2}-{1-i\sqrt{2}\over 1^2+(\sqrt{2})^2}=$$

$$={3+i3\sqrt{2}-1+i\sqrt{2}\over 3}={2+i4\sqrt{2}\over 3}$$

$P(\dfrac{4}{3},\dfrac{2\sqrt{2}}{3}), Q(\dfrac{2}{3},\dfrac{4\sqrt{2}}{3}),O(0,0)$

$x_M=\dfrac{x_P+x_Q}{2}=\dfrac{\dfrac{4}{3}+\dfrac{2}{3}}{2}=1,$

$y_M=\dfrac{y_P+y_Q}{2}=\dfrac{\dfrac{2\sqrt{2}}{3}+\dfrac{4\sqrt{2}}{3}}{2}=\sqrt{2}$

$M(1, \sqrt{2})$

$$\lambda={OG\over GM}=2$$

$x_G=\dfrac{x_0+2x_M}{1+2}=\dfrac{0+2\cdot1}{3}=\dfrac{2}{3},$

$y_G=\dfrac{y_0+2y_M}{1+2}=\dfrac{0+2\cdot\sqrt{2}}{3}=\dfrac{2\sqrt{2}}{3},$

$G(\dfrac{2}{3},\dfrac{2\sqrt{2}}{3})$

$$\overline{GP}=(x_P-x_G,y_P-y_G)=({2\over 3}, 0)$$ $$\overline{GQ}=(x_Q-x_G,y_Q-y_G)=(0, {2\sqrt{2}\over 3})$$

$$|\overline{GP}|={2\over 3}\not=0,|\overline{GQ}|={2\sqrt{2}\over 3}\not=0$$

$\overline{GQ}\cdot\overline{GP}=0=>GQ\perp GP$

Hence  angle PGQ is a right angle.