Question

x^2+y^2=9
16x^2-4y^2=64

Answer 1

are you supposed to solve for x and y?

Answer 2

Solving for x^2 and y^2, using linear algebra:
u = x^2, v = y^2

(i) u + v = 9
(ii) 3u - 2v = 32 [dividing your second equation by 2]
(iii) 2u + 2v = 18 [doubling equation (i)]
5u = 50 [adding equations (ii) and (iii)]
u = 10
10 + v = 9
v = -1.

Then x = +- sqrt(10), y = +-i.

If you were to graph the circle and the hyperbola in the same x-y plane, you would see that they do not intersect anywhere.