To determine the equation of a line that is perpendicular to a given line and passes through a specific point, we can follow these steps:
1. Identify the slope of the given line: The given equation is [tex]\( y - 3x - 2 = 0 \)[/tex]. We need to rearrange this equation into slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
[tex]\[
y - 3x - 2 = 0 \implies y = 3x + 2
\][/tex]
Thus, the slope [tex]\( m \)[/tex] of the given line is [tex]\( 3 \)[/tex].
2. Determine the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line will be:
[tex]\[
m_{\perpendicular} = -\frac{1}{3}
\][/tex]
3. Use the point-slope form to find the equation: The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1) = (6, 8)\)[/tex] and [tex]\( m = -\frac{1}{3} \)[/tex].
Substituting these values into the point-slope form yields:
[tex]\[
y - 8 = -\frac{1}{3}(x - 6)
\][/tex]
4. Simplify to get the standard form of the line: First, distribute the slope on the right-hand side:
[tex]\[
y - 8 = -\frac{1}{3}x + 2
\][/tex]
Then, add [tex]\( 8 \)[/tex] to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{3}x + 2 + 8
\][/tex]
[tex]\[
y = -\frac{1}{3}x + 10
\][/tex]
5. Select the correct answer: The equation of the line that is perpendicular to [tex]\( y - 3x - 2 = 0 \)[/tex] and passes through the point [tex]\( (6, 8) \)[/tex] is:
[tex]\[
y = -\frac{1}{3}x + 10
\][/tex]
Thus, the correct answer is [tex]\(\boxed{A}\)[/tex].
