Dear all, can you please help me show why the 2nd scenario below has no solution......(adsbygoogle = window.adsbygoogle || []).push({});

Theorem 3.3.1 Suppose that a and b are not both zero. Then the equation ## aZ + b\bar Z + c = 0 ##

has

a, b and c are complex numbers. Z is the unknown complex number.

(1) a unique solution if and only if ## |a| != |b| ##;

(2) no solution if and only if |a| = |b| and ## b\bar c != c\bar a ##;

(3) a line of solutions if and only if |a| = |b| and ## b\bar c = c\bar a ##.

Answer:

(1) a unique solution

In the equation ## aZ + b\bar Z + c = 0 ##

a,b and c are complex numbers.

for |a| = |b|, say ##a = f + ig##

then |a| = ## \sqrt{f^2 + g^2} ##

negative f and g will be equal to positive because of the square root

so b can be ## f + ig, f - ig, -f + ig, -f -ig, g + if, g - if, -g + if, -g - if ##

if |a| and |b| are equal i.e. when a and b are any of the complex numbers above then there may be repeated solutions for z.

if ##z = z1 + iz2 ## then for two different a and b in the above list z1 will same and z2 will be same.

(2) no solution

if |a| = |b| then ONLY repeated solutions will occur and a and b are only one of the eight complex numbers given, c is any complex number???

what does the condition ## b\bar c != c\bar a## do?

(3) line of solutions

(3) what does the condition ## b\bar c = c\bar a## do?

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# Doubt about the solutions of the equation aZ + bz + c = 0

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