4.
Figure 2
The curve $C_{1}$ with parametric equations
\[
x=10 \cos t, \quad y=4 \sqrt{2} \sin t, \quad 0 \leqslant t< 2 \pi
\]
meets the circle $C_{2}$ with equation
\[
x^{2}+y^{2}=66
\]
at four distinct points as shown in Figure 2.
Given that one of these points, $S$, lies in the 4 th quadrant, find the Cartesian coordinates of $S$.
(6)
Answer
The intersection point S in the 4th quadrant has coordinates: \(\boxed{(7.07106781186548, -4.00000000000000)}\)
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: \(\boxed{(7.07106781186548, -4.00000000000000)}\)
