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De-Moivre's Theorem

If z=x+iy s...

Question

If $z = x + i y$ satisfies arg $(z + 2)$ $=$ arg $(z + i)$ , then? find relation between real and imaginary part of z.

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Solution

Given,

$z = x + i y$

$a r g (z + 2) = a r g (z + i)$

$a r g (x + 2 + i y) = a r g (x + i (y + 1))$

${tan}^{- 1} (\frac{y}{x + 2}) = {tan}^{- 1} (\frac{y + 1}{x})$

taking $tan$ on both sides, we get,

$\frac{y}{x + 2} = \frac{y + 1}{x}$

$x y = (x + 2) (y + 1)$

$x y = x y + x + 2 y + 2$

$∴ x + 2 y + 2 = 0$

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