x^2 + y^2 = 9
y^2=9-x^2
x^2/4 + y^2/36 = 1
x^2/4 + (9-x^2)/36 = 1
9x^2+9-x^2=36
8x^2=27
x=+-sqrt(27/8) so there are two values of x where the lines meet. BUT
y = +-sqrt(9-x^2) so there are two values of y for each value of x where the lines meet.
2*2=4
Or you can recognize that the first is a circle of diameter 6 around (0,0) and the second is an elipse 4 wide by 12 high around (0,0). 12 is bigger than 6 and 6 is bigger than 4 so the two overlap and meet at four points.