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Algebraic Equation

If z1 , z2 ar...

Question

If z1 , z2 are two complex numbers such that arg(z1+z2) = 0 andIm(z1z2) = 0, then
(A) z1 = - z2
(B) z1 = z2
(C) z 1= z2
(D) none of these

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Solution

$Let the two complex numbers are, z_{1} = a + ib z_{2} = x + iy So \arg (z_{1} + z_{2}) = \arg (a + x + i (b + y)) = 0 \tan^{- 1} (\frac{b + y}{a + x}) = 0 \frac{b + y}{a + x} = 0 b + y = 0 . . . . . (1) Now Im (z_{1} z_{2}) = Im (ax - by + i (ay + bx)) = 0 (ay + bx) = 0 ay = - bx Now using equation 1, ay = xy a = x So we get b = - y and a = x So i = out complex number are, z_{1} = x - iy z_{2} = x + iy From here we can say that |z_{1}| = |z_{2}| = \sqrt{x^{2} + y^{2}}$

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