33ok.. khoj lete hain..
Q.1
If |z|=1
then prove that complex number denoted by √(1+z)/(1-z) always lie in either of the two fixed perpendicular lines..
Q.2
If z1+z2+z3=0. then prove that the angle between any two pair of complex number is greater than equal to 2Ï€/3
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8 Answers
62Q 1 was solved sometime back on the forum.. I think it was given by eureka.. i will have to check..
62http://targetiit.com/iit_jee_forum/posts/complex_geometry_295.html
341Q 1.
z lies on a circle for which the line joining (1,0) and (-1,0) are the ends of a diameter.
That means arg(1+z/1-z) = ±π/2.
Hence the argument of the square root of the number is ± π/4. Thus, the numbers lie on one of two perpendicular lines.
Q 2. Needs to be rephrased I guess. It should be: there exists a pair such that the angle between them is greater than or equal to 2Ï€/3.
Please check
62this proof for the first part is awesome :)
I wonder how much I have to learn to equal prophet :)
62Rotation as done by Prophet is a graphical method if u understand it well enuf :)