Key Concepts and Formulas

• Midpoint Formula: The midpoint $M$ of a line segment joining two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

• Slope Formula: The slope $m$ of a line passing through two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Perpendicular Lines: If two non-vertical lines are perpendicular, the product of their slopes is $-1$. That is, if $m_1$ is the slope of the first line and $m_2$ is the slope of the second line, then: $m_1 \cdot m_2 = -1$

• Equation of a Line (Point-Slope Form): The equation of a line with slope $m$ passing through a point $(x_0, y_0)$ is given by: $y - y_0 = m(x - x_0)$

Step-by-Step Solution

Let the given points be $P(1, 4)$ and $Q(k, 3)$.

Step 1: Find the Midpoint of the Line Segment PQ.

• Why: The perpendicular bisector passes through the midpoint of the segment PQ.
• Action: Apply the midpoint formula using $P(1, 4)$ as $(x_1, y_1)$ and $Q(k, 3)$ as $(x_2, y_2)$. $M = \left(\frac{1 + k}{2}, \frac{4 + 3}{2}\right) = \left(\frac{1 + k}{2}, \frac{7}{2}\right)$ Let's denote the coordinates of the midpoint as $(x_M, y_M) = \left(\frac{1 + k}{2}, \frac{7}{2}\right)$.

Step 2: Calculate the Slope of the Line Segment PQ.

• Why: We need the slope of PQ to find the slope of the line perpendicular to it.
• Action: Apply the slope formula for points $P(1, 4)$ and $Q(k, 3)$. $m_{PQ} = \frac{3 - 4}{k - 1} = \frac{-1}{k - 1}$

Step 3: Determine the Slope of the Perpendicular Bisector.

• Why: The perpendicular bisector is perpendicular to PQ, so we use the negative reciprocal relationship of their slopes.
• Action: Let $m_{PB}$ be the slope of the perpendicular bisector. $m_{PB} = -\frac{1}{m_{PQ}} = -\frac{1}{\left(\frac{-1}{k - 1}\right)} = k - 1$

Step 4: Formulate the Equation of the Perpendicular Bisector.

• Why: We have a point (the midpoint $M$) that the bisector passes through and its slope ($m_{PB}$). We can use the point-slope form to write its equation.
• Action: Substitute $M\left(\frac{1 + k}{2}, \frac{7}{2}\right)$ as $(x_0, y_0)$ and $m_{PB} = k-1$ into the point-slope form $y - y_0 = m(x - x_0)$. $y - \frac{7}{2} = (k - 1)\left(x - \frac{1 + k}{2}\right)$

Step 5: Use the Y-intercept Information to Solve for k.

• Why: The problem states that the y-intercept of the perpendicular bisector is -4. This means the line passes through the point $(0, -4)$. We can substitute these coordinates into the line's equation to find $k$.
• Action: Substitute $x=0$ and $y=-4$ into the equation from Step 4. $-4 - \frac{7}{2} = (k - 1)\left(0 - \frac{1 + k}{2}\right)$ Combine the terms on the left side: $\frac{-8 - 7}{2} = (k - 1)\left(-\frac{1 + k}{2}\right)$ $\frac{-15}{2} = -\frac{(k - 1)(k + 1)}{2}$ Multiply both sides by 2: $-15 = -(k - 1)(k + 1)$ Multiply both sides by -1: $15 = (k - 1)(k + 1)$ Recognize the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$: $15 = k^2 - 1$ Add 1 to both sides: $k^2 = 16$ Take the square root of both sides: $k = \pm 4$

Step 6: Check the Options.

• Why: We found two possible values for $k$, but the question asks for "a possible value" and provides multiple-choice options.
• Action: The possible values for $k$ are $4$ and $-4$. Comparing these with the given options: (A) 1 (B) 2 (C) -2 (D) -4 The value $k = -4$ matches option (D). However, the correct answer is $k=1$. Let's re-examine.

Step 5 (Revised): Use the Y-intercept Information to Solve for k.

• Why: The problem states that the y-intercept of the perpendicular bisector is -4. This means the line passes through the point $(0, -4)$. We can substitute these coordinates into the line's equation to find $k$.
• Action: Substitute $x=0$ and $y=-4$ into the equation from Step 4. $-4 - \frac{7}{2} = (k - 1)\left(0 - \frac{1 + k}{2}\right)$ Combine the terms on the left side: $\frac{-8 - 7}{2} = (k - 1)\left(-\frac{1 + k}{2}\right)$ $\frac{-15}{2} = -\frac{(k - 1)(k + 1)}{2}$ Multiply both sides by 2: $-15 = -(k - 1)(k + 1)$ Multiply both sides by -1: $15 = (k - 1)(k + 1)$ Recognize the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$: $15 = k^2 - 1$ Add 1 to both sides: $k^2 = 16$ Take the square root of both sides: $k = \pm 4$

Step 6 (Revised): Check the Options.

• Why: We found two possible values for $k$, but the question asks for "a possible value" and provides multiple-choice options. We made an error in the calculation. Let's revisit Step 5. The error lies in assuming k=-4 is the answer. Let's substitute k=1 into the equation and see if it works. The slope of PQ is then undefined. PQ is a vertical line x=1. The midpoint is (1, 7/2). The perpendicular bisector is a horizontal line y = 7/2. This does NOT have a y-intercept of -4. So k=1 is not the answer.

Let's re-examine Step 5. $-4 - \frac{7}{2} = (k - 1)\left(0 - \frac{1 + k}{2}\right)$ $\frac{-15}{2} = (k-1)\left(-\frac{k+1}{2}\right)$ $-15 = -(k^2-1)$ $15 = k^2 - 1$ $k^2 = 16$ $k = \pm 4$

The options are 1, 2, -2, -4. k = -4 is an option. Let's test k=1.

If the answer is 1, m = (3-4)/(1-1) which is undefined. The midpoint is (1, 7/2). The perpendicular bisector is y=7/2, which does not have a y-intercept of -4. If the answer is -4, m = (3-4)/(-4-1) = 1/5. The midpoint is (-3/2, 7/2). The slope of the perpendicular bisector is -5. The line is y - 7/2 = -5(x + 3/2). y = -5x -15/2 + 7/2 = -5x - 8/2 = -5x - 4. The y-intercept is -4. If k=2, m = (3-4)/(2-1) = -1. Midpoint = (3/2, 7/2). Perpendicular slope = 1. y - 7/2 = 1(x - 3/2). y = x - 3/2 + 7/2 = x + 4/2 = x+2. y-intercept = 2. If k=-2, m = (3-4)/(-2-1) = 1/3. Midpoint = (-1/2, 7/2). Perpendicular slope = -3. y - 7/2 = -3(x + 1/2). y = -3x -3/2 + 7/2 = -3x + 4/2 = -3x + 2. y-intercept = 2. There must be an error in the answer key.

Common Mistakes & Tips

• Double-check algebraic manipulations, especially when dealing with fractions and negative signs.
• Remember the definition of the y-intercept: the value of y when x=0.
• Always verify your solution by plugging the found value(s) back into the original equation(s) to ensure consistency.

Summary

After careful step-by-step calculation and verification, we found that $k = -4$ satisfies the given conditions. This corresponds to option (D). However, the correct answer according to the problem is A (k=1). I believe there is an error in the answer key. The correct answer should be -4.

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