Key Concepts and Formulas

• Centroid of a Triangle: The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by the coordinates: $C = \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \right)$
• Intersection of Two Lines: To find the point of intersection of two lines, we solve their equations simultaneously.
• Equation of a Line: Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the line passing through them is: $\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$ Equivalently, a point $(x, y)$ lies on the line if and only if the equation holds.

Step-by-Step Solution

Step 1: Find the coordinates of the centroid C.

We are given the vertices of the triangle as (3, –1), (1, 3), and (2, 4). Using the centroid formula:

$C = \left( \frac{3+1+2}{3}, \frac{-1+3+4}{3} \right) = \left( \frac{6}{3}, \frac{6}{3} \right) = (2, 2)$

Therefore, the centroid C has coordinates (2, 2).

Step 2: Find the coordinates of the point of intersection P.

We are given the equations of the lines as $x + 3y - 1 = 0$ and $3x - y + 1 = 0$. We need to solve these equations simultaneously. From the first equation, we have $x = 1 - 3y$. Substituting this into the second equation gives:

$3(1 - 3y) - y + 1 = 0$ $3 - 9y - y + 1 = 0$ $4 - 10y = 0$ $10y = 4$ $y = \frac{4}{10} = \frac{2}{5}$

Now, substitute $y = \frac{2}{5}$ back into $x = 1 - 3y$:

$x = 1 - 3\left(\frac{2}{5}\right) = 1 - \frac{6}{5} = \frac{5 - 6}{5} = -\frac{1}{5}$

Therefore, the point of intersection P has coordinates $\left(-\frac{1}{5}, \frac{2}{5}\right)$.

Step 3: Find the equation of the line passing through C and P.

We have C(2, 2) and P$\left(-\frac{1}{5}, \frac{2}{5}\right)$. Using the equation of a line passing through two points:

$\frac{y - 2}{x - 2} = \frac{\frac{2}{5} - 2}{-\frac{1}{5} - 2} = \frac{\frac{2 - 10}{5}}{\frac{-1 - 10}{5}} = \frac{-8}{-11} = \frac{8}{11}$

So, $\frac{y - 2}{x - 2} = \frac{8}{11}$

$11(y - 2) = 8(x - 2)$ $11y - 22 = 8x - 16$ $8x - 11y + 6 = 0$

Step 4: Check which of the given points lies on the line $8x - 11y + 6 = 0$.

(A) (–9, –7): $8(-9) - 11(-7) + 6 = -72 + 77 + 6 = 11 \neq 0$

(B) (9, 7): $8(9) - 11(7) + 6 = 72 - 77 + 6 = 1 \neq 0$

(C) (7, 6): $8(7) - 11(6) + 6 = 56 - 66 + 6 = -4 \neq 0$

(D) (–9, –6): $8(-9) - 11(-6) + 6 = -72 + 66 + 6 = 0$

Upon closer inspection, there's an arithmetic error in option (C). It should be: (C) (7, 6): $8(7) - 11(6) + 6 = 56 - 66 + 6 = -4$. Let's check option (D) again. (D) (-9, -6): $8(-9) - 11(-6) + 6 = -72 + 66 + 6 = 0$.

There seems to be an error in the provided answer. Let's re-evaluate our steps to ensure accuracy.

Step 1 (Revisited): Centroid Calculation $C = \left( \frac{3+1+2}{3}, \frac{-1+3+4}{3} \right) = \left( \frac{6}{3}, \frac{6}{3} \right) = (2, 2)$. This is correct.

Step 2 (Revisited): Intersection Point $x + 3y - 1 = 0 \implies x = 1 - 3y$ $3x - y + 1 = 0 \implies 3(1-3y) - y + 1 = 0 \implies 3 - 9y - y + 1 = 0 \implies 4 - 10y = 0 \implies y = \frac{2}{5}$ $x = 1 - 3(\frac{2}{5}) = 1 - \frac{6}{5} = -\frac{1}{5}$. This is also correct. $P = (-\frac{1}{5}, \frac{2}{5})$

Step 3 (Revisited): Equation of the Line $\frac{y - 2}{x - 2} = \frac{\frac{2}{5} - 2}{-\frac{1}{5} - 2} = \frac{\frac{-8}{5}}{\frac{-11}{5}} = \frac{8}{11}$ $11(y - 2) = 8(x - 2) \implies 11y - 22 = 8x - 16 \implies 8x - 11y + 6 = 0$. This is still correct.

Step 4 (Revisited): Checking the points (A) (-9, -7): $8(-9) - 11(-7) + 6 = -72 + 77 + 6 = 11 \neq 0$ (B) (9, 7): $8(9) - 11(7) + 6 = 72 - 77 + 6 = 1 \neq 0$ (C) (7, 6): $8(7) - 11(6) + 6 = 56 - 66 + 6 = -4 \neq 0$. This is where the error was. (D) (-9, -6): $8(-9) - 11(-6) + 6 = -72 + 66 + 6 = 0$.

It appears the correct answer is indeed (D). The provided "Correct Answer: C" is wrong.

Common Mistakes & Tips

• Always double-check your arithmetic, especially when dealing with fractions and negative signs.
• When solving systems of equations, ensure you substitute correctly to avoid errors.
• After finding the equation of the line, verify your answer by plugging in the coordinates of the points used to define the line.
• If the given answer doesn't match your solution, re-examine each step carefully.

Summary

We found the centroid C of the triangle and the point of intersection P of the two lines. Then, we determined the equation of the line passing through C and P. Finally, we checked which of the given points satisfied this equation. The point (-9, -6) is the one that satisfies the equation $8x - 11y + 6 = 0$.

Final Answer

The final answer is \boxed{(-9, -6)}, which corresponds to option (D).