Key Concepts and Formulas

• Slope of a Line: Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ of the line passing through them is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Perpendicular Lines: If two lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1 m_2 = -1$.
• Midpoint Formula: The midpoint of a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Equation of a Line (Slope-Intercept Form): The equation of a line with slope $m$ and y-intercept $c$ is $y = mx + c$.

Step-by-Step Solution

Step 1: Find the slope of line segment PQ. The slope of the line segment joining P(1, 4) and Q(k, 3) is given by: $m_{PQ} = \frac{3 - 4}{k - 1} = \frac{-1}{k - 1}$

Step 2: Find the slope of the perpendicular bisector. Since the perpendicular bisector is perpendicular to PQ, its slope ($m_{\perp}$) is the negative reciprocal of $m_{PQ}$: $m_{\perp} = -\frac{1}{m_{PQ}} = -\frac{1}{\frac{-1}{k-1}} = k - 1$

Step 3: Find the midpoint of line segment PQ. The midpoint M of the line segment joining P(1, 4) and Q(k, 3) is: $M = \left(\frac{1 + k}{2}, \frac{4 + 3}{2}\right) = \left(\frac{1 + k}{2}, \frac{7}{2}\right)$

Step 4: Find the equation of the perpendicular bisector. We know the slope of the perpendicular bisector is $k - 1$ and it passes through the midpoint $M\left(\frac{1 + k}{2}, \frac{7}{2}\right)$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, we get: $y - \frac{7}{2} = (k - 1)\left(x - \frac{1 + k}{2}\right)$

Step 5: Use the given y-intercept to find k. The y-intercept is the point where the line crosses the y-axis, which means x = 0. We are given that the y-intercept is -4. So, substitute x = 0 and y = -4 into the equation of the perpendicular bisector: $-4 - \frac{7}{2} = (k - 1)\left(0 - \frac{1 + k}{2}\right)$ $-\frac{15}{2} = (k - 1)\left(-\frac{1 + k}{2}\right)$ Multiply both sides by -2: $15 = (k - 1)(k + 1)$ $15 = k^2 - 1$ $k^2 = 16$ $k = \pm 4$

Step 6: Re-examine the question and options. We have two possible values for k, $k = 4$ or $k = -4$. Let's check which one satisfies the equation.

If $k = 4$, then $m_{PQ} = \frac{-1}{4-1} = -\frac{1}{3}$ and $m_{\perp} = 4-1 = 3$. The midpoint is $(\frac{5}{2}, \frac{7}{2})$. The equation of the perpendicular bisector is $y - \frac{7}{2} = 3(x - \frac{5}{2})$. Setting x = 0 gives $y = \frac{7}{2} - \frac{15}{2} = -\frac{8}{2} = -4$. So $k=4$ is a valid solution.

If $k = -4$, then $m_{PQ} = \frac{-1}{-4-1} = \frac{1}{5}$ and $m_{\perp} = -4-1 = -5$. The midpoint is $(\frac{-3}{2}, \frac{7}{2})$. The equation of the perpendicular bisector is $y - \frac{7}{2} = -5(x + \frac{3}{2})$. Setting x = 0 gives $y = \frac{7}{2} - \frac{15}{2} = -\frac{8}{2} = -4$. So $k=-4$ is also a valid solution.

However, only $k = -4$ appears as an option. The question asks for "a value of k", so we choose the available option.

Step 7: Check the options again.

Since $k^2 = 16$, we have $k = \pm 4$. The options are $\sqrt{14}$, -4, -2, and $\sqrt{15}$. Thus, $k = -4$ is a valid answer.

Common Mistakes & Tips

• Remember that the slope of a perpendicular line is the negative reciprocal.
• Be careful with signs when substituting values into equations.
• Always double-check your calculations to avoid simple arithmetic errors.
• Read the question carefully and make sure you are answering what is being asked. In this case, the question asks for a value of k, not all values of k.

Summary

We found the slope of the line segment PQ, then determined the slope of its perpendicular bisector. Using the midpoint formula, we found the coordinates of the midpoint of PQ, which lies on the perpendicular bisector. We then used the point-slope form to find the equation of the perpendicular bisector. Finally, we used the given y-intercept to solve for k. The possible values for k are $\pm 4$. Among the given options, only -4 is present.

The final answer is \boxed{-4}, which corresponds to option (B).