Two circles $S_1: x^2 + y^2 - 16 = 0$ and $S_2: x^2 + y^2 - 6x - 8y - 8 = 0$ intersect at points $A$ and $B$. Find the coordinates of point $P$ on the common chord.
Step-by-Step Solution
Key Concept: The common chord of two circles is found by subtracting their equations; this gives a line equation that passes through both intersection points.
The common chord is found by subtracting the two circle equations: $S_1 - S_2$ gives $3x + 4y - 4 = 0$, which is identical to the chord of contact $AB$. Using the ratio of distances from centers, we get $\frac{r_1}{r_2} = \frac{15}{8}$. Solving the system with the chord equation and the circle equations yields the intersection point. The coordinates of $P$ on the common chord are $(12, 16)$.
Correct Answer: (12, 16)