If (-2,6) is the image of the point (4,2) with respect to line L=0, then L is:

Explanation

To find the equation of the line \( L \) given that the point \((-2, 6)\) is the image of the point \((4, 2)\) with respect to the line \( L \), we can follow these steps:

### Step 1: Identify the Points
Let \( P(4, 2) \) be the original point and \( Q(-2, 6) \) be its image with respect to the line \( L \).

### Step 2: Find the Midpoint
The midpoint \( M \) of the segment \( PQ \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( P \) and \( Q \):
\[ M = \left( \frac{4 + (-2)}{2}, \frac{2 + 6}{2} \right) = \left( \frac{2}{2}, \frac{8}{2} \right) = (1, 4) \]

### Step 3: Calculate the Slope of Line \( PQ \)
The slope \( m_{PQ} \) of the line segment \( PQ \) is given by:
\[ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 2}{-2 - 4} = \frac{4}{-6} = -\frac{2}{3} \]

### Step 4: Determine the Slope of Line \( L \)
Since line \( L \) is the mirror line, it is perpendicular to line \( PQ \). The product of the slopes of two perpendicular lines is \(-1\). Therefore:
\[ m_{PQ} \cdot m_L = -1 \implies -\frac{2}{3} \cdot m_L = -1 \]
Solving for \( m_L \):
\[ m_L = \frac{3}{2} \]

### Step 5: Write the Equation of Line \( L \)
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) = (1, 4) \) and \( m = \frac{3}{2} \):
\[ y - 4 = \frac{3}{2}(x - 1) \]
Expanding this:
\[ y - 4 = \frac{3}{2}x - \frac{3}{2} \]
Rearranging gives:
\[ y = \frac{3}{2}x + \frac{5}{2} \]
To express this in standard form:
\[ 3x - 2y + 5 = 0 \]

### Final Answer
The equation of the line \( L \) is:
\[ 3x - 2y + 5 = 0 \]

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