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Complex Numbers with a Property (TOMATO Subjective 88)

Problem: A pair of complex numbers \(z_1, z_2\) is said to have the property \(P\) if for every complex number \(z\) we find real numbers \(r\) and \(s\) such that \(z=rz_1 + sz_2\).Show that a pair of complex numbers has property \(P\) if and only if the points \(z_1,z_2\) and \(0\) on the complex plane are not collinear.

Solution: Let the complex numbers \(z_1,z_2,0\) be collinear, and the line joining them make an angle \(\theta\) with the x-axis. This means that:

\(arg(z_1) =arg(z_2) = \theta\)

\(=> z_1 = |z_1| (cos\,\theta + i sin\, \theta)\)

Similarly,

\(=> z_2 = |z_2| (cos\,\theta + i sin\, \theta)\)

Therefore, \(z=rz_1 + sz_2\)

\(=> z =r |z_1| (cos\,\theta + i sin\, \theta) + s|z_2| (cos\,\theta + i sin\, \theta)\)

\(=> z =(r |z_1| + s|z_2|) (cos\,\theta + i sin\, \theta)\)

Which implies that \(z\) lies on the same line that joins \(z_1\) and \(z_2\). But that is not true, as \(z\) can be any complex number.

Thus the assumption that \(z_1, z_2, 0\) are collinear is false.

Hence Proved.

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