If z1 and z2 are complex numbers satisfying the equations \left|\right. z \left|\right. = \sqrt{5} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \left|\right. \frac{z - \sqrt{2} \left(\right. 1 + i \left.\right)}{z} \left|\right. = 1 \textrm{ } then find |z1

Engineering

Mathematics

Geometry of Complex Number

Question

If z₁ and z₂ are complex numbers satisfying the equations $| z | = \sqrt{5} and | \frac{z - \sqrt{2} (1 + i)}{z} | = 1$ then find |z₁ – z₂|.

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$∵ | \frac{z - \sqrt{2} (1 + i)}{z} | = 1$

$\Rightarrow | z - \sqrt{2} (1 + i) | = | z | = | z - 0 |$

$∴$ z is perpendicular bisectors of line joining $(\sqrt{2}, \sqrt{2})$ and (0, 0) where (0, 0) is centre of $| z | = \sqrt{5}$ with radius $\sqrt{5}$ .

Clearly OM = 1 and OA = $\sqrt{5}$

$∴$ AM = 2

$∴$ AB = |z₁ – z₂| = 4