Let's solve the problem step by step.

Given:

• Coordinates of $A(2, 6)$, $B(6, 1)$, and $C(p, q)$.
• $B$ is the midpoint of $AC$.
• We need to find the equation of the line passing through $C(p, q)$ and perpendicular to line $AB$, in the form $ax + by = c$.

Step 1: Find the coordinates of $C(p, q)$

Since $B(6, 1)$ is the midpoint of $AC$, we can use the midpoint formula to express the coordinates of $C$.

Midpoint formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ where $A(2, 6)$ and $C(p, q)$ are the points, and $B(6, 1)$ is the midpoint.

For the x-coordinate: $\frac{2 + p}{2} = 6 \implies 2 + p = 12 \implies p = 10$

For the y-coordinate: $\frac{6 + q}{2} = 1 \implies 6 + q = 2 \implies q = -4$

Thus, $C(10, -4)$.

Step 2: Find the slope of line $AB$

The slope of line $AB$ is given by: $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 6}{6 - 2} = \frac{-5}{4}$

Step 3: Find the slope of the line perpendicular to $AB$

The slope of the line perpendicular to $AB$ is the negative reciprocal of $m_{AB}$: $m_{\text{perpendicular}} = \frac{4}{5}$

Step 4: Find the equation of the line passing through $C(10, -4)$ and perpendicular to $AB$

Using the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$ where $m = \frac{4}{5}$, and $(x_1, y_1) = (10, -4)$.

Substitute the values: $y + 4 = \frac{4}{5}(x - 10)$ Multiply both sides by 5 to eliminate the fraction: $5(y + 4) = 4(x - 10)$ Simplify: $5y + 20 = 4x - 40$ Rearranging the terms: $4x - 5y = 60$

Thus, the equation of the line is: $4x - 5y = 60$

Final Answer:

The equation of the line is $4x - 5y = 60$.

Would you like more details or have any other questions? Here are 5 related questions:

1. How do you find the equation of a line given two points?
2. What is the midpoint formula and how can it be applied in other problems?
3. How do you determine if two lines are perpendicular?
4. Can you explain how to convert a point-slope form to standard form?
5. How do you solve a system of equations involving two lines?

Tip: Always check if the problem involves perpendicular or parallel lines, as this gives valuable information about their slopes!