Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Test of Mathematics Solution Subjective 88 - Complex Numbers with a Property

Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 88 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.


Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

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.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com