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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.

October 16, 2016