Let z1,z2 and z3 be the vertices of a triangle in a complex plane and |z1|=x,|z2|=y and |z3|=z such that x3+y3+z3=3xyz, then arg(z3z2)=k arg(z3−z1z2−z1) where k is equal to
Question
Let z1,z2 and z3 be the vertices of a triangle in a complex plane and |z1|=x,|z2|=y and |z3|=z such that x3+y3+z3=3xyz, then arg(z3z2)=k arg(z3−z1z2−z1) where k is equal to
Solution
The problem is asking for the value of k in the given equation. This is a complex number problem involving the argument of complex numbers and the modulus of complex numbers.
The given condition x³ + y³ + z³ = 3xyz is a special case of the identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca), which holds true if a + b + c = 0.
So, we can say that |z1| + |z2| + |z3| = x + y + z = 0.
Since the modulus of a complex number cannot be negative, the only way this can hold true is if x = y = z = 0.
This implies that z1, z2, and z3 are all at the origin of the complex plane.
The argument of a complex number at the origin is undefined, so the equation arg(z3z2) = k arg((z3 - z1)/(z2 - z1)) does not have a defined value for k.
Therefore, the problem does not have a solution in the set of real numbers.
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