Step 1 :Substitute the parametric equations of $C_1$ into the equation of $C_2$ and solve for $t$: \(32.0\sin^2(t) + 100\cos^2(t) = 66\)
Step 2 :Find the Cartesian coordinates of the intersection points by plugging the values of $t$ back into the parametric equations of $C_1$: \(x = 10\cos(t)\), \(y = 5.65685424949238\sin(t)\)
Step 3 :The intersection point S in the 4th quadrant has coordinates: \(\boxed{(7.07106781186548, -4.00000000000000)}\)