If |z1|=|z2| and argz1~argz2=pi,show that z1+z2=0?

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#argz1~argz2=pi#

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Answer 1 · 2017-12-30T07:27:30

Please refer to the Explanation.

Suppose that, #|z_1|=|z_2|=r >0, &, argz_2=theta in (-pi,pi]#.

Since, #argz_1-argz_2=pi,# we have, #argz_1=pi+theta#.

Consequently, #z_2=r(costheta+isintheta), and, #

#z_1=r(cos(pi+theta)+isin(pi+theta)), or, #

#z_1=r(-costheta-isintheta)#,

#=-z_2#.

# rArr z_1+z_2=0#

Q.E.D.

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