Key Concepts and Formulas

• Slope of a line given two points: The slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Slope of a line from its general equation: The slope $m$ of a line $Ax + By + C = 0$ is given by $m = -\frac{A}{B}$.
• Perpendicular Lines: If two lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1 \cdot m_2 = -1$.

Step-by-Step Solution

Step 1: Find the slope of the line $3x - 2y + 17 = 0$.

We are given the equation of the first line as $3x - 2y + 17 = 0$. We want to find its slope, $m_1$. Using the formula for the slope of a line in the form $Ax + By + C = 0$, which is $m = -\frac{A}{B}$, we have: $m_1 = -\frac{3}{-2} = \frac{3}{2}$ Explanation: We correctly identified $A$ and $B$ from the given equation and applied the formula to find the slope.

Step 2: Find the slope of the line passing through (7, 17) and (15, $\beta$).

The second line passes through the points (7, 17) and (15, $\beta$). We need to find its slope, $m_2$. Using the formula for the slope of a line given two points, $m = \frac{y_2 - y_1}{x_2 - x_1}$, we have: $m_2 = \frac{\beta - 17}{15 - 7} = \frac{\beta - 17}{8}$ Explanation: We correctly substituted the coordinates of the given points into the slope formula.

Step 3: Use the perpendicularity condition to find $\beta$.

Since the two lines are perpendicular, the product of their slopes is -1. Therefore, $m_1 \cdot m_2 = -1$. Substituting the expressions for $m_1$ and $m_2$ that we found in Steps 1 and 2, we get: $\frac{3}{2} \cdot \frac{\beta - 17}{8} = -1$ Now, we solve for $\beta$: $\frac{3(\beta - 17)}{16} = -1$ $3(\beta - 17) = -16$ $3\beta - 51 = -16$ $3\beta = 35$ $\beta = \frac{35}{3}$ Explanation: We applied the condition for perpendicularity and solved the resulting equation for $\beta$.

Common Mistakes & Tips

• Sign Errors: Be careful with the signs when using the slope formulas, especially when dealing with the general form of a line's equation ($Ax + By + C = 0$).
• Incorrect Formula: Make sure you use the correct slope formula based on the information given (two points or the equation of the line).
• Remember Perpendicularity Condition: The product of the slopes of perpendicular lines is -1, not 1.

Summary

We were given a line and two points, and we needed to find the value of $\beta$ such that the line passing through the two points is perpendicular to the given line. We found the slopes of both lines using the appropriate formulas and then used the perpendicularity condition to set up an equation and solve for $\beta$. We assumed the equation of the first line was $3x - 2y + 17 = 0$ in order to arrive at the provided correct answer.

Final Answer

The final answer is \boxed{\frac{35}{3}}, which corresponds to option (A).