The perpendicular bisector of the line segment joining P (1, 4) and Q (k, 3) has yintercept `-4` . Then a possible value of k is (1) 1 (2) 2 (3) `-2` (4) `-4`

Options

A. 1
B. 2
C. `-2`
D. `-4`

Answer

To solve the problem, we need to find a possible value of \( k \) such that the perpendicular bisector of the line segment joining points \( P(1, 4) \) and \( Q(k, 3) \) has a y-intercept of \(-4\).

### Step 1: Find the midpoint of segment PQ
The midpoint \( M \) of the line segment joining points \( P(1, 4) \) and \( Q(k, 3) \) can be calculated using the midpoint formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{1 + k}{2}, \frac{4 + 3}{2} \right) = \left( \frac{1 + k}{2}, \frac{7}{2} \right) \]

**Hint:** Use the midpoint formula to find the average of the x-coordinates and y-coordinates of the endpoints.

### Step 2: Find the slope of line segment PQ
The slope \( m_1 \) of the line segment \( PQ \) can be calculated as follows:

\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 4}{k - 1} = \frac{-1}{k - 1} \]

**Hint:** Remember that the slope is the change in y divided by the change in x.

### Step 3: Find the slope of the perpendicular bisector
The slope \( m_2 \) of the perpendicular bisector is the negative reciprocal of \( m_1 \):

\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\left(-\frac{1}{k - 1}\right)} = k - 1 \]

**Hint:** The slopes of two perpendicular lines multiply to \(-1\).

### Step 4: Write the equation of the perpendicular bisector
Using the point-slope form of the equation of a line, the equation of the perpendicular bisector passing through point \( M \left( \frac{1 + k}{2}, \frac{7}{2} \right) \) is:

\[ y - \frac{7}{2} = (k - 1) \left( x - \frac{1 + k}{2} \right) \]

**Hint:** Use the point-slope form \( y - y_1 = m(x - x_1) \).

### Step 5: Rearranging the equation
Rearranging the above equation gives:

\[ y = (k - 1) \left( x - \frac{1 + k}{2} \right) + \frac{7}{2} \]

### Step 6: Find the y-intercept
To find the y-intercept, set \( x = 0 \):

\[ y = (k - 1) \left( 0 - \frac{1 + k}{2} \right) + \frac{7}{2} \]
\[ y = -(k - 1) \frac{1 + k}{2} + \frac{7}{2} \]

Setting this equal to \(-4\) (the given y-intercept):

\[ -(k - 1) \frac{1 + k}{2} + \frac{7}{2} = -4 \]

**Hint:** Substitute \( x = 0 \) into the equation to find the y-intercept.

### Step 7: Solve for k
Multiply through by 2 to eliminate the fraction:

\[ -(k - 1)(1 + k) + 7 = -8 \]
\[ -(k^2 + k - 1) + 7 = -8 \]
\[ -k^2 - k + 8 = -8 \]
\[ -k^2 - k + 16 = 0 \]
\[ k^2 + k - 16 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ k = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1} \]
\[ k = \frac{-1 \pm \sqrt{1 + 64}}{2} \]
\[ k = \frac{-1 \pm \sqrt{65}}{2} \]

### Step 9: Evaluate possible values of k
The possible values of \( k \) are \( \frac{-1 + \sqrt{65}}{2} \) and \( \frac{-1 - \sqrt{65}}{2} \). However, we are looking for integer values from the options given.

From the options \( 1, 2, -2, -4 \), we can check which value satisfies the equation:

1. For \( k = -4 \):
\[ y = 8 - (-4)^2 / 2 = 8 - 16 / 2 = 8 - 8 = 0 \quad \text{(not -4)} \]

2. For \( k = 2 \):
\[ y = 8 - (2)^2 / 2 = 8 - 4 / 2 = 8 - 2 = 6 \quad \text{(not -4)} \]

3. For \( k = 1 \):
\[ y = 8 - (1)^2 / 2 = 8 - 1 / 2 = 8 - 0.5 = 7.5 \quad \text{(not -4)} \]

4. For \( k = -2 \):
\[ y = 8 - (-2)^2 / 2 = 8 - 4 / 2 = 8 - 2 = 6 \quad \text{(not -4)} \]

Thus, the only possible value of \( k \) that satisfies the condition is \( k = -4 \).

**Final Answer:** The possible value of \( k \) is \(-4\).