The perpendicular bisector of the line segment joining A(1, 4) and B(t, 3) has y - intercept equal to `-4`. Then, the product of all possible values of t is equal to

Options

A. 1
B. 2
C. `-16`
D. `-4`

Answer

To solve the problem, we need to find the product of all possible values of \( t \) such that the perpendicular bisector of the line segment joining points \( A(1, 4) \) and \( B(t, 3) \) has a y-intercept of \(-4\).

### Step 1: Find the midpoint \( C \) of segment \( AB \)

The coordinates of the midpoint \( C \) can be calculated using the midpoint formula:
\[ C\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( A(1, 4) \) and \( B(t, 3) \).

Thus,
\[ C = \left( \frac{1 + t}{2}, \frac{4 + 3}{2} \right) = \left( \frac{1 + t}{2}, \frac{7}{2} \right) \]

**Hint:** Use the midpoint formula to find the coordinates of the midpoint between two points.

### Step 2: Find the slope \( m_{AB} \) of line segment \( AB \)

The slope \( m_{AB} \) is given by:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 4}{t - 1} = \frac{-1}{t - 1} \]

**Hint:** Remember that the slope of a line is the change in \( y \) divided by the change in \( x \).

### Step 3: Find the slope \( m_{PQ} \) of the perpendicular bisector

Since the perpendicular bisector is perpendicular to \( AB \), the slope \( m_{PQ} \) is the negative reciprocal of \( m_{AB} \):
\[ m_{PQ} = -\frac{1}{m_{AB}} = -\frac{1}{\left(-\frac{1}{t-1}\right)} = t - 1 \]

**Hint:** The product of the slopes of two perpendicular lines is \(-1\).

### Step 4: Write the equation of the perpendicular bisector \( PQ \)

Using the point-slope form of the equation of a line, we have:
\[ y - y_1 = m(x - x_1) \]
Substituting \( C\left( \frac{1+t}{2}, \frac{7}{2} \right) \) and \( m_{PQ} = t - 1 \):
\[ y - \frac{7}{2} = (t - 1)\left(x - \frac{1+t}{2}\right) \]

**Hint:** Use the point-slope form to write the equation of a line given a point and a slope.

### Step 5: Find the y-intercept of the line

To find the y-intercept, set \( x = 0 \):
\[ y - \frac{7}{2} = (t - 1)\left(0 - \frac{1+t}{2}\right) \]
This simplifies to:
\[ y - \frac{7}{2} = -(t - 1) \cdot \frac{1+t}{2} \]
\[ y = \frac{7}{2} - \frac{(t - 1)(1 + t)}{2} \]

Setting \( y = -4 \) (the given y-intercept):
\[ -4 = \frac{7}{2} - \frac{(t - 1)(1 + t)}{2} \]

**Hint:** To find the y-intercept, substitute \( x = 0 \) into the line equation.

### Step 6: Solve for \( t \)

Multiply through by 2 to eliminate the fraction:
\[ -8 = 7 - (t - 1)(1 + t) \]
Rearranging gives:
\[ (t - 1)(1 + t) = 15 \]
Expanding:
\[ t^2 - 1 = 15 \]
\[ t^2 - 16 = 0 \]
Factoring:
\[ (t - 4)(t + 4) = 0 \]

Thus, the possible values of \( t \) are \( t = 4 \) and \( t = -4 \).

**Hint:** Factor the quadratic equation to find the possible values of \( t \).

### Step 7: Find the product of all possible values of \( t \)

The product of the values \( t = 4 \) and \( t = -4 \) is:
\[ 4 \times (-4) = -16 \]

**Final Answer:** The product of all possible values of \( t \) is \(-16\).