Given:
C1:∣z∣=4,C2:z+z1
Here, ∣z∣=4 is a circle x2+y2=16.
Now, let z=4eiθ,
So, z+z1=4eiθ+4e−iθ
⇒x+iy=4cosθ+i4sinθ+4cosθ−4isinθtaking z+z1=x+iy
Now on comparing both side we get,
x=\frac{17}{4}\mathrm{cos}\theta &y=\frac{15}{4}\mathrm{sin}\theta
Now on solving cos2θ+sin2θ=1 we get,
(417)2x2+(415)2y2=1
Which is a equation of ellipse,

Therefore, curves C1 and C2 intersect at 4 points.