To determine the equation of a line that is perpendicular to the line [tex]\( y = 2x + 4 \)[/tex] and passes through the point [tex]\( (4, 6) \)[/tex], we need to follow these steps:
### Step 1: Find the slope of the given line
The given line is [tex]\( y = 2x + 4 \)[/tex]. The slope of this line (in the format [tex]\( y = mx + b \)[/tex]) is [tex]\( m = 2 \)[/tex].
### Step 2: Determine the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope [tex]\( m_{\text{perp}} \)[/tex] of the perpendicular line is:
[tex]\[
m_{\text{perp}} = -\frac{1}{2}
\][/tex]
### Step 3: Use the point-slope form to find the equation of the perpendicular line
The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line, and [tex]\( m \)[/tex] is the slope of the line.
Using the point [tex]\( (4, 6) \)[/tex] and the slope [tex]\( -\frac{1}{2} \)[/tex], we plug these values into the point-slope form:
[tex]\[
y - 6 = -\frac{1}{2}(x - 4)
\][/tex]
### Step 4: Simplify the equation
First, distribute the slope on the right-hand side:
[tex]\[
y - 6 = -\frac{1}{2}x + 2
\][/tex]
Next, add 6 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{2}x + 2 + 6
\][/tex]
[tex]\[
y = -\frac{1}{2}x + 8
\][/tex]
### Step 5: Match the equation with the given options
The equation [tex]\( y = -\frac{1}{2}x + 8 \)[/tex] corresponds to option A.
Therefore, the equation of the line that is perpendicular to [tex]\( y = 2x + 4 \)[/tex] and passes through the point [tex]\( (4, 6) \)[/tex] is:
[tex]\[
\boxed{y = -\frac{1}{2}x + 8}
\][/tex]
