Answer to Question #124662 in Geometry for desmond

Question #124662
) If z1 = 1 + 2i, find the set of values of z2 for which
(i) |z1 + z2| = |z1| + |z2| (ii) |z1 + z2| = |z1| − |z2|
1
Expert's answer
2020-07-20T18:40:11-0400

Given z1=1+2iz_1 = 1 + 2i and let z2=x+iyz_2 = x+iy .

(i) We have z1+z2=z1+z2|z_1 + z_2| = |z_1| + |z_2|

    (1+x)+i(2+y)=1+2i+x+iy\implies |(1+x)+i(2+y)| = |1+2i|+|x+iy|

    (1+x)2+(2+y)2=5+x2+y2\implies \sqrt{(1+x)^2+(2+y)^2}= \sqrt{5}+\sqrt{x^2+y^2}

By squaring on both sides, we getx2+y2+2x+4y+5=5+x2+y2+25x2+5y2x^2+y^2+2x+4y+5 = 5 + x^2+y^2 +2\sqrt{5x^2+5y^2}

    x+2y=5x2+5y2\implies x+2y = \sqrt{5x^2+5y^2}

Again by squaring on both side, we get x2+4y2+4xy=5x2+5y2x^2+4y^2 + 4xy = 5x^2+5y^2

    4x2+y24xy=0\implies 4x^2 + y^2 - 4xy = 0

    (2xy)2=0    2xy=0    y=2x\implies(2x-y)^2 = 0 \\ \implies 2x- y = 0 \implies y = 2x .

Hence, z2=x(1+2i)z_2 = x(1+2i) .


(ii) We have z1+z2=z1z2|z_1 + z_2| = |z_1| - |z_2|

    (1+x)+i(2+y)=1+2ix+iy\implies |(1+x)+i(2+y)| = |1+2i|-|x+iy|

    (1+x)2+(2+y)2=5x2+y2\implies \sqrt{(1+x)^2+(2+y)^2}= \sqrt{5}-\sqrt{x^2+y^2}

By squaring on both sides, we get

x2+y2+2x+4y+5=5+x2+y225x2+5y2x^2+y^2+2x+4y+5 = 5 + x^2+y^2 -2\sqrt{5x^2+5y^2}

    x+2y=5x2+5y2\implies x+2y = -\sqrt{5x^2+5y^2}

Again by squaring on both side, we get x2+4y2+4xy=5x2+5y2x^2+4y^2 + 4xy = 5x^2+5y^2

    4x2+y24xy=0\implies 4x^2 + y^2 - 4xy = 0

    (2xy)2=0    2xy=0    y=2x\implies(2x-y)^2 = 0 \\ \implies 2x- y = 0 \implies y = 2x .

Hence, z2=x(1+2i)z_2 = x(1+2i) .


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