The number of complex numbers $z$ that satisfies both the equation $| z - 1 - i | = \sqrt{2}$ and $| z + 1 + i | = 2$ , is

1

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2

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4

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infinite

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Solution

The correct option is B $2$
$| z - 1 - i | = \sqrt{2}$
This represents a circle with centre, $C_{1} (1, 1)$ and radius $r_{1} = \sqrt{2}$

$| z + 1 + i | = 2$
This represents a circle with centre, $C_{2} (- 1, - 1)$ and radius $r_{2} = 2$

Now, $C_{1} C_{2} = \sqrt{2^{2} + 2^{2}} = 2 \sqrt{2}$
$r_{2} + r_{1} = 2 + \sqrt{2} r_{2} - r_{1} = 2 - \sqrt{2}$
So, $r_{2} + r_{1} > C_{1} C_{2} > r_{2} - r_{1}$
Therefore, these two circles intersect at two points only.
Hence, there are two possible complex numbers satisfying the given equations.

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