1. **State the problem:** We are given two circles $C_1$ and $C_2$ intersecting at points $A$ and $B$. The equation of $C_1$ is $$x^2 + y^2 - 16x - 10y + 39 = 0,$$ and points $A$ and $B$ lie on the line $$3x - y = -1.$$ Circle $C_2$ passes through point $(-13, 2)$. We want to find the coordinates of points $A$ and $B$ and understand the relationship between the circles and the line. 2. **Rewrite the line equation:** The line can be expressed as $$y = 3x + 1.$$ This will help us substitute $y$ in the circle equation. 3. **Substitute $y$ into $C_1$'s equation:** Replace $y$ with $3x + 1$ in the circle equation: $$x^2 + (3x + 1)^2 - 16x - 10(3x + 1) + 39 = 0.$$ 4. **Expand and simplify:** $$x^2 + 9x^2 + 6x + 1 - 16x - 30x - 10 + 39 = 0,$$ which simplifies to $$10x^2 - 40x + 30 = 0.$$ 5. **Divide through by 10:** $$x^2 - 4x + 3 = 0.$$ 6. **Factor the quadratic:** $$(x - 3)(x - 1) = 0,$$ so $$x = 3 \quad \text{or} \quad x = 1.$$ 7. **Find corresponding $y$ values:** Using $y = 3x + 1$, - For $x=1$, $y = 3(1) + 1 = 4$, - For $x=3$, $y = 3(3) + 1 = 10$. 8. **Coordinates of intersection points:** $$A = (1, 4), \quad B = (3, 10).$$ 9. **Summary:** Points $A$ and $B$ lie on both circle $C_1$ and the line $3x - y = -1$. Circle $C_2$ also passes through these points and the point $(-13, 2)$, but its full equation is not given here. **Final answer:** Points of intersection $A = (1, 4)$ and $B = (3, 10)$.