can we say that if |z1|= |z2|= |z3| and z1 + z2 + z3 = 0
one of z1 ,z2,z3 is real?
is it obvious that z1, z2,z3 are the roots of x3= |z1|3
they ll be the vertices of an equilateral triangle but are above things always true?
pls explain !!
49draw the figure..
convert this problem to a problem in coordinate geometry and then this will become clear to you!! :)
21i hav done this question, i further want to analyse it;
actually the question was if |Z1| = |Z2| = |z3| and z1 + z2 + z3 = 0 prove they are vertices of equilateral triangle:
i have done this part
i want knw if we can conclude that at least one of them is real or if they are the roots of x3= |z1|3
341You cannot conclude whether they are real or not. But they will certainly form the vertices of an equilateral triangle
Let |z1|=k.
Now, z_1+z_2+z_3=0
Also, \overline{z_1}+\overline{z_2}+\overline{z_3}=0 \Rightarrow k^2 \left(\frac{1}{z_1} + \frac{1}{z_2}+\frac{1}{z_3}\right)=0 \Rightarrow \frac{1}{z_1} + \frac{1}{z_2}+\frac{1}{z_3} = 0
So, z1, z2 and z3 are roots of z^3-q = 0 where q is a complex number.
Let q = k e^{i \theta}
Then the three roots i.e. the three vertices are
k^{\frac{1}{3}} e^{i \frac{\theta}{3}}, \omega k^{\frac{1}{3}} e^{i \frac{\theta}{3}}, \omega^2 k^{\frac{1}{3}} e^{i \frac{\theta}{3}}
Its now obvious that the three vertices form an equilateral triangle. But none of them need be real.
21there is one more way to say they forms vertices of equilateral triangle
its clear that z1,z2,z3 lie on a circle,the circumcenter of the triangle formed by z1,z2,z3 is the origin
further we have z1 + z2 + z3 = 0
so the centroid is also the origin
so it is equilateral triangle;)
@THEPROPHET: sir,what abt the hint given in the question i uploaded? how can we assume that one of them is real without any loss of generality?
341WLOG You can assume z1 is real, because multiplying each of the numbers by e^{i \theta} does not alter any of the conditions. Geometrically, you are rotating the figure by \theta. So you can rotate it appropriately so that one of the numbers become real. In fact you may assume it to be a positive real number.