Question:
<p>Let C be the centroid of the triangle with vertices (3, -1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y - 1 = 0 and 3x - y + 1 = 0. Then the line passing through the points C and P also passes through the point:</p>
<p style="display:inline">(-9, -6)</p>
<p style="display:inline">(-9, -7)</p>
<p style="display:inline">(9, 7)</p>
<p style="display:inline">(7, 6)</p>
Step-by-Step Solution
Key Concept: Solve for the centroid and point of intersection to determine the specific line equation using the two-point formula.
<p>Coordinates of centroides<br />
C = <span class="math-tex">$\left(\frac{x_{1}+x_{2}+x_{3}}{3}, \frac{y_{1}+y_{2}+y_{3}}{3}\right)$</span><br />
= <span class="math-tex">$\left(\frac{3+1+2}{3}, \frac{-1+3+4}{3}\right)$</span> = (2, 2)<br />
The given equation of lines are<br />
x + 3y - 1 = 0 ...(i)<br />
3x - y + 1 = 0 ...(ii)<br />
Then, from (i) and (ii)<br />
point of intersection P<span class="math-tex">$\left(-\frac{1}{5}, \frac{2}{5}\right)$</span><br />
equation of line DP<br />
8x - 11y + 6 = 0</p>
Correct Answer: A