To solve the problem, let's take the following steps:
1. Identify the slope of the original line:
The given equation of the line is \( y = -\frac{1}{2} x + 11 \).
From this equation, we can determine that the slope \( m \) of this line is \( -\frac{1}{2} \).
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
Therefore, the slope \( m_{\perpendicular} \) of the perpendicular line is:
[tex]\[ m_{\perpendicular} = -\frac{1}{-1/2} = 2 \][/tex]
3. Use the point-slope form equation:
The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, we have the slope \( m_{\perpendicular} = 2 \) and the point \((x_1, y_1) = (4, -8)\).
Plug in these values into the point-slope form:
[tex]\[ y - (-8) = 2(x - 4) \][/tex]
Simplify the left side:
[tex]\[ y + 8 = 2(x - 4) \][/tex]
4. Match the equation to the given options:
We need to identify which of the provided options matches the equation \( y + 8 = 2(x - 4) \).
Comparing with the given options:
- A. \( y-4=2(x+8) \) does not match.
- B. \( y+8=\frac{1}{2}(x-4) \) does not match.
- C. \( y-8=\frac{1}{2}(x+4) \) does not match.
- D. \( y+8=2(x-4) \) matches the form we derived.
Thus, the correct equation in point-slope form for the line perpendicular to \( y = -\frac{1}{2} x + 11 \) that passes through \( (4, -8) \) is:
[tex]\[ \boxed{y + 8 = 2(x - 4)} \][/tex]
