Add the two equations together to find y, then use that to find x.

(x^2 + y^2) + (y^2 - x^2) = (25) + (7)

2y^2 = 32 (collect terms)

y^2 = 16 (divide by 2)

y = +/- sqrt(16) = +/- 4

x^2 + 16 = 25 (substitute for y^2 in the first equation)

x^2 = 9 (subtract 16)

x = +/- sqrt(9) = +/- 3

The points of intersection of the circle and hyperbola are

(3, 4), (3, -4), (-3, -4), (-3, 4)

(x^2 + y^2) + (y^2 - x^2) = (25) + (7)

2y^2 = 32 (collect terms)

y^2 = 16 (divide by 2)

y = +/- sqrt(16) = +/- 4

x^2 + 16 = 25 (substitute for y^2 in the first equation)

x^2 = 9 (subtract 16)

x = +/- sqrt(9) = +/- 3

The points of intersection of the circle and hyperbola are

(3, 4), (3, -4), (-3, -4), (-3, 4)