To determine the equation of the line that is perpendicular to the given line and passes through the point [tex]\((2, 6)\)[/tex], let's proceed step-by-step:
1. Identify the nature of the given line:
The given line is [tex]\(x = 2\)[/tex]. This is a vertical line because in this equation, the x-coordinate is always 2, regardless of the y-coordinate.
2. Understand perpendicular lines:
A line perpendicular to a vertical line must be horizontal. This is because a vertical line has an undefined slope, and the slope of a perpendicular line (in this case) must be the negative reciprocal, which translates to a zero slope. A line with a zero slope is horizontal.
3. Equation of a horizontal line:
The equation of a horizontal line takes the form [tex]\(y = c\)[/tex], where [tex]\(c\)[/tex] is the y-coordinate that the line passes through.
4. Determine the y-coordinate:
The line must pass through the point [tex]\((2, 6)\)[/tex]. Since it is horizontal, it will maintain the same y-coordinate for all values of x.
5. Form the equation:
Given that the point [tex]\((2, 6)\)[/tex] has a y-coordinate of 6, the equation of the horizontal line passing through this point is simply:
[tex]\[
y = 6
\][/tex]
Therefore, the equation of the line that is perpendicular to the given vertical line [tex]\(x = 2\)[/tex] and passes through the point [tex]\((2, 6)\)[/tex] is [tex]\(y = 6\)[/tex].
Thus, the correct answer is:
[tex]\[
y = 6
\][/tex]
