Question #116820
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
1
Expert's answer
2020-05-21T17:54:14-0400

p=z+1z=1+i2+11+i2p=z+\frac1z=1+i\sqrt2+\frac1{1+i\sqrt2}

Multiply the numerator and denominator of the fraction by 1i21-i\sqrt2

p=1+i2+1i23=43+223ip=1+i\sqrt2+\frac{1-i\sqrt2}{3}=\frac43+\frac{2\sqrt2}{3}i


Similarly, we do with qq


q=z1z=1+i211+i2=1+i21i23=23+423iq=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 PP has coordinates (43\frac43, 223\frac{2\sqrt2}{3}) , and QQ (23\frac23, 423\frac{4\sqrt2}{3}) . Find the coordinates MM


Mx=Px+Qx2=1M_x=\frac{P_x+Q_x}2=1My=Py+Qy2=2M_y=\frac{P_y+Q_y}2=\sqrt2

Find the coordinates GG


Gx=Mx23=23G_x=M_x\cdot\frac23=\frac23

Gy=My23=223G_y=M_y\cdot\frac23=\frac{2\sqrt2}3


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

GPx=PxGx=23GP_x=P_x-G_x=\frac23

GPy=PyGy=0GP_y=P_y-G_y=0

GQx=QxGx=0GQ_x=Q_x-G_x=0


GQx=QyGy=223GQ_x=Q_y-G_y=\frac{2 \sqrt2}3

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



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