Key Concepts and Formulas

• Euler Line: The orthocentre ($H$), centroid ($G$), and circumcentre ($O$) of a triangle are collinear, and this line is called the Euler line.
• Centroid Division: The centroid ($G$) divides the line segment joining the orthocentre ($H$) and the circumcentre ($O$) in the ratio $2:1$, i.e., $HG:GO = 2:1$.
• Distance Formula (Parallel to a Line): The distance between a point $(x_1, y_1)$ and a line $ax + by + c = 0$ measured parallel to the line $y = mx + d$ is given by $\left| \frac{ax_1 + by_1 + c}{a+bm} \right| \sqrt{1+m^2}$.

Step-by-Step Solution

Step 1: Apply the Centroid Division Formula

Since the centroid $G(a, b)$ divides the line segment joining the orthocentre $H(-6, -8)$ and the circumcentre $O(3, 4)$ in the ratio $2:1$, we can use the section formula: $a = \frac{2(3) + 1(-6)}{2+1} = \frac{6-6}{3} = 0$ $b = \frac{2(4) + 1(-8)}{2+1} = \frac{8-8}{3} = 0$ Therefore, $A = G(0, 0)$.

Step 2: Determine the Coordinates of Point P

Given that $P(2a+3, 7b+5)$ and $a = 0$ and $b = 0$, we have: $P(2(0)+3, 7(0)+5) = P(3, 5)$

Step 3: Calculate the Slope of the Parallel Line

The distance is measured parallel to the line $x - 2y - 1 = 0$. We rewrite this line in slope-intercept form to find its slope: $2y = x - 1$ $y = \frac{1}{2}x - \frac{1}{2}$ The slope of this line is $m = \frac{1}{2}$.

Step 4: Apply the Distance Formula (Parallel to a Line)

We need to find the distance of the point $P(3, 5)$ from the line $2x + 3y - 4 = 0$ measured parallel to the line $x - 2y - 1 = 0$. Using the distance formula with $x_1 = 3$, $y_1 = 5$, $a = 2$, $b = 3$, $c = -4$, and $m = \frac{1}{2}$: $\text{Distance} = \left| \frac{2(3) + 3(5) - 4}{2 + 3(\frac{1}{2})} \right| \sqrt{1 + \left(\frac{1}{2}\right)^2}$ $\text{Distance} = \left| \frac{6 + 15 - 4}{2 + \frac{3}{2}} \right| \sqrt{1 + \frac{1}{4}}$ $\text{Distance} = \left| \frac{17}{\frac{7}{2}} \right| \sqrt{\frac{5}{4}}$ $\text{Distance} = \left| \frac{34}{7} \right| \frac{\sqrt{5}}{2}$ $\text{Distance} = \frac{34}{7} \cdot \frac{\sqrt{5}}{2}$ $\text{Distance} = \frac{17 \sqrt{5}}{7}$

Common Mistakes & Tips

• Remember the correct ratio for the Euler line: $HG:GO = 2:1$. Confusing this ratio will lead to incorrect coordinates for the centroid.
• Be careful when substituting values into the distance formula, especially the slope of the parallel line.
• Double-check your arithmetic to avoid errors in the calculations.

Summary

We used the properties of the Euler line and the centroid to determine the coordinates of the centroid. Then we found the coordinates of point $P$. Finally, we applied the formula for the distance between a point and a line, measured parallel to another line, to find the desired distance. The distance of the point $P(3, 5)$ from the line $2x + 3y - 4 = 0$ measured parallel to the line $x - 2y - 1 = 0$ is $\frac{17\sqrt{5}}{7}$.

Final Answer

The final answer is $\boxed{\frac{17 \sqrt{5}}{7}}$, which corresponds to option (C).