>>12521903Roots are pairs of integers 0<=a,b<=p-1 such that a^2=b^3+7 modulo p, i.e. such that p divides a^2-b^3-7. 2^200 is gigantic, so there are plenty roots.
If you mean why the line intersects another point, (ignoring the special cases) you just need to write the equation of the line y=cx+d and insert it into y^2=x^3+7. This is a polynomial of degree in one variable with two roots over the field. So by basic algebra all three roots are in the field.
