Question #117384
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 PQ and G is the point on OM such that OG =
2 3
OM. Prove that angle PGQ is a right angle.
1
Expert's answer
2020-05-21T17:35:37-0400
z=1+i2z=1+i\sqrt{2}

p=z+1z=1+i2+11+i2=1+i2+1i212+(2)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+i32+1i23=4+i223={3+i3\sqrt{2}+1-i\sqrt{2}\over 3}={4+i2\sqrt{2}\over 3}

q=z1z=1+i211+i2=1+i21i212+(2)2=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+i321+i23=2+i423={3+i3\sqrt{2}-1+i\sqrt{2}\over 3}={2+i4\sqrt{2}\over 3}

P(43,223),Q(23,423),O(0,0)P(\dfrac{4}{3},\dfrac{2\sqrt{2}}{3}), Q(\dfrac{2}{3},\dfrac{4\sqrt{2}}{3}),O(0,0)


xM=xP+xQ2=43+232=1,x_M=\dfrac{x_P+x_Q}{2}=\dfrac{\dfrac{4}{3}+\dfrac{2}{3}}{2}=1,


yM=yP+yQ2=223+4232=2y_M=\dfrac{y_P+y_Q}{2}=\dfrac{\dfrac{2\sqrt{2}}{3}+\dfrac{4\sqrt{2}}{3}}{2}=\sqrt{2}


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


λ=OGGM=2\lambda={OG\over GM}=2

xG=x0+2xM1+2=0+213=23,x_G=\dfrac{x_0+2x_M}{1+2}=\dfrac{0+2\cdot1}{3}=\dfrac{2}{3},


yG=y0+2yM1+2=0+223=223,y_G=\dfrac{y_0+2y_M}{1+2}=\dfrac{0+2\cdot\sqrt{2}}{3}=\dfrac{2\sqrt{2}}{3},


G(23,223)G(\dfrac{2}{3},\dfrac{2\sqrt{2}}{3})


GP=(xPxG,yPyG)=(23,0)\overline{GP}=(x_P-x_G,y_P-y_G)=({2\over 3}, 0)

GQ=(xQxG,yQyG)=(0,223)\overline{GQ}=(x_Q-x_G,y_Q-y_G)=(0, {2\sqrt{2}\over 3})

GP=230,GQ=2230|\overline{GP}|={2\over 3}\not=0,|\overline{GQ}|={2\sqrt{2}\over 3}\not=0


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

Hence angle PGQ is a right angle.


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