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+z1=1+i2+1+i21
Multiply the numerator and denominator of the fraction by 1−i2
p=1+i2+31−i2=34+322i
Similarly, we do with q
q=z−z1=1+i2−1+i21=1+i2−31−i2=32+342i
In this diagram, the point P has coordinates (34, 322) , and Q (32, 342) . Find the coordinates M
Mx=2Px+Qx=1My=2Py+Qy=2
Find the coordinates G
Gx=Mx⋅32=32
Gy=My⋅32=322
We define two vectors GP(GPx,GPy) and GQ(GQx,GQy) , if their scalar product is 0, then the angle between them is 90°
GPx=Px−Gx=32
GPy=Py−Gy=0
GQx=Qx−Gx=0
GQx=Qy−Gy=322
GP⋅GQ=GPx⋅GQx+GPy⋅GQy=0 ------> ∠PGQ is a right angle
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