Question

The number of complex numbers $z_1$ which can simultaneously satisfy both the equations $|$z - 2$|$ = 2 and z(1 - i) + $\overline{z}(1+i)$ = 4 is equal to

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Answer 1

First equation represents a circle and the second one represents a straight line (we will express it in terms of x and y later). A straight line and circle can have zero, one or two points in common. To determine the number of common points, we will need radius of the circle and the perpendicular distance of the center from the straight line. In this case won't calculate them.(It will be clear once you find the equation of the straight line)
Let Z = x + iy.
$|$z - 2 $|$ = 2 is a circle with center (2,0) and radius 2.
$z(1-i)+\overline{z}(1+i)$ = 4 can be written as
$z+\overline{z}+i(\overline{z}-z)$ = 4
$\Rightarrow 2x+i(-2iy)$ = 4
$\Rightarrow 2x+2y$ = 4
$\Rightarrow x+y$ = 2
x + y = 2 is a line passing through the center (2,0) of the circle.
Or it is the diameter. So it will cut the circle at two points.