Question

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

• A. $1$
• B. $2$
• C. $4$
• D. infinite

Answer 1

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.