Abstract: In elliptic curve theory, number of rational points on
elliptic curves and determination of these points is a fairly important
problem. Let p be a prime and Fp be a finite field and k ∈ Fp. It
is well known that which points the curve y2 = x3 + kx has and
the number of rational points of on Fp. Consider the circle family
x2 + y2 = r2. It can be interesting to determine common points of
these two curve families and to find the number of these common
points. In this work we study this problem.