My algebra has failed me!

Algebra Level 4

Consider two complex numbers w w and z z . Suppose that these two numbers satisfy

{ ∣ w ∣ = ∣ z ∣ ∣ w + z ∣ = ∣ w − z ∣ \begin{cases} |w|=|z| \\ |w+z|=|w-z| \end{cases}

Then this implies that w = ± k × z w=\pm k \times z .

What is ∣ Im ( k ) ∣ + ∣ Re ( k ) ∣ |\text{Im}(k)|+|\text{Re}(k)| ?


The answer is 1.

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Dylan Pentland by Dylan Pentland , 21, USA

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1 solution

When | w + z| = |w - z|, it geometrically means that w is equidistant from z and -z. So w lies on the perpendicular bisector to the line joining z and -z. Also |z| = |w|. So w is z rotated by pi/2 (90 degrees). (This could be negative also). So w = +/ - iz. Clearly the answer is 1.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly what I had in mind! Nice solution :)

Dylan Pentland - 5 years, 8 months ago

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