The problem involves points $A(9, -1)$ and $B(1, 7)$ in the Cartesian plane. Point $C$ lies on the line segment $AB$ and satisfies the condition $|AC| = 3|BC|$. We are tasked with finding the coordinates of point $C$.

Step 1: Parametric representation of point $C$

Since $C$ lies on the line segment $AB$, it can be represented parametrically as: $C(x, y) = (1-t)A + tB$ Where $t \in [0, 1]$ and: $C(x, y) = ((1-t) \cdot 9 + t \cdot 1, (1-t) \cdot -1 + t \cdot 7)$ Simplify the coordinates: $C(x, y) = (9 - 8t, -1 + 8t)$

Step 2: Condition $|AC| = 3|BC|$

The condition $|AC| = 3|BC|$ implies that the distances satisfy: $|AC|^2 = 9|BC|^2$ Using the distance formula, compute $|AC|^2$ and $|BC|^2$: $|AC|^2 = (x_C - x_A)^2 + (y_C - y_A)^2$ $|BC|^2 = (x_C - x_B)^2 + (y_C - y_B)^2$

Substituting $A(9, -1)$, $B(1, 7)$, and $C(9 - 8t, -1 + 8t)$: $|AC|^2 = ((9 - 8t) - 9)^2 + ((-1 + 8t) - (-1))^2 = (-8t)^2 + (8t)^2 = 64t^2 + 64t^2 = 128t^2$ $|BC|^2 = ((9 - 8t) - 1)^2 + ((-1 + 8t) - 7)^2 = (8 - 8t)^2 + (-8 + 8t)^2 = (64 - 128t + 64t^2) + (64 - 128t + 64t^2)$ $|BC|^2 = 128 - 256t + 128t^2$

Using the condition $|AC|^2 = 9|BC|^2$: $128t^2 = 9(128 - 256t + 128t^2)$ Simplify: $128t^2 = 1152 - 2304t + 1152t^2$ $0 = 1152 - 2304t + 1024t^2$ Divide by 64: $0 = 18 - 36t + 16t^2$ Rearrange into standard quadratic form: $16t^2 - 36t + 18 = 0$ Simplify further: $8t^2 - 18t + 9 = 0$

Step 3: Solve the quadratic equation

Using the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 8$, $b = -18$, and $c = 9$: $t = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(8)(9)}}{2(8)}$ $t = \frac{18 \pm \sqrt{324 - 288}}{16}$ $t = \frac{18 \pm \sqrt{36}}{16}$ $t = \frac{18 \pm 6}{16}$ $t = \frac{24}{16} = \frac{3}{2} \quad \text{or} \quad t = \frac{12}{16} = \frac{3}{4}$

Since $t \in [0, 1]$, we take $t = \frac{3}{4}$.

Step 4: Find coordinates of $C$

Substitute $t = \frac{3}{4}$ into $C(x, y) = (9 - 8t, -1 + 8t)$: $x_C = 9 - 8 \cdot \frac{3}{4} = 9 - 6 = 3$ $y_C = -1 + 8 \cdot \frac{3}{4} = -1 + 6 = 5$

Thus, the coordinates of $C$ are: $C(3, 5)$

Do you want further clarifications or details on any step? Here are some related questions:

1. How is the parametric representation of a point on a line derived?
2. Can we verify the solution by rechecking distances?
3. What if the condition was $|AC| = 2|BC|$? How would it change the result?
4. How can this problem be extended to three dimensions?
5. Is there a geometric interpretation of $|AC| = 3|BC|$?

Tip: When solving problems involving ratios, parametric equations are often the most efficient method!