Find the locus of points of the plane which satisfy the following equations.
$| z + i | = | z + 2 | .$

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Solution

Let $z = x + i y$

Then, $| x + i y + i | = | x + i y + 2 |$

$⟹ \sqrt{x^{2} + (y + 1)^{2}} = \sqrt{(x + 2)^{2} + y^{2}}$

$⟹ x^{2} + y^{2} + 2 y + 1 = x^{2} + 4 x + 4 + y^{2}$ ........[squaring both sides]

$⟹ 4 x - 2 y + 3 = 0$ ......[equation of a straight line]

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∣ ∣ ∣ \frac{z + i}{z - 3 i} ∣ ∣ ∣ = 1.

| z - 2 | = | z + 2 i |

| z - 1 | + | z - i | = \sqrt{2}

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| i z - 1 | + | z - i | = 2

| z + i - 2 | ⩽ 2

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