The reflection of the point `(t - 1,2t+ 2)` in a line is `(2t + 1,t)`, then the equation of the line has slope equals to

Options

A. `-1`
B. ` 1`
C. ` 1/2 `
D. ` 0 `

Answer

To find the slope of the line given the reflection of the point \((t - 1, 2t + 2)\) is \((2t + 1, t)\), we can follow these steps:

### Step 1: Identify the Points
Let point A be \((t - 1, 2t + 2)\) and point B be \((2t + 1, t)\).

### Step 2: Find the Slope of Line AB
The slope \(m_2\) of line segment AB can be calculated using the formula:
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
- \(x_1 = t - 1\), \(y_1 = 2t + 2\)
- \(x_2 = 2t + 1\), \(y_2 = t\)

So,
\[ m_2 = \frac{t - (2t + 2)}{(2t + 1) - (t - 1)} \]

### Step 3: Simplify the Slope Calculation
Calculating the numerator:
\[ t - (2t + 2) = t - 2t - 2 = -t - 2 \]

Calculating the denominator:
\[ (2t + 1) - (t - 1) = 2t + 1 - t + 1 = t + 2 \]

Thus, the slope \(m_2\) becomes:
\[ m_2 = \frac{-t - 2}{t + 2} \]

### Step 4: Relationship Between Slopes
Since line AB is perpendicular to line L (the line of reflection), we have:
\[ m_1 \cdot m_2 = -1 \]
where \(m_1\) is the slope of line L.

### Step 5: Substitute and Solve for \(m_1\)
Substituting \(m_2\) into the equation:
\[ m_1 \cdot \frac{-t - 2}{t + 2} = -1 \]

Rearranging gives:
\[ m_1 = \frac{-1 \cdot (t + 2)}{-t - 2} = \frac{t + 2}{t + 2} = 1 \]

### Conclusion
Thus, the slope of the line L is:
\[ \boxed{1} \]