To determine the equation of the line that is perpendicular to the given line [tex]\(x = 2\)[/tex] and passes through the point [tex]\((2, 6)\)[/tex], follow these steps:
1. Understand the Given Line:
- The given line equation is [tex]\(x = 2\)[/tex]. This is a vertical line where all points on the line have the x-coordinate equal to 2, regardless of the y-coordinate.
2. Determine the Perpendicular Line:
- A line perpendicular to a vertical line is a horizontal line. Vertical lines have undefined slopes, and the slopes of perpendicular lines in this case go from undefined (vertical) to zero (horizontal).
3. Use the Given Point:
- We need the perpendicular line to pass through the point [tex]\((2, 6)\)[/tex].
4. Form the Equation:
- Since the line we are seeking is horizontal and passes through [tex]\((2, 6)\)[/tex], the y-coordinate will remain constant at 6 for all points on the line.
- This implies the equation for the line can be written as [tex]\(y = 6\)[/tex].
Therefore, the equation of the line that is perpendicular to [tex]\(x=2\)[/tex] and passes through the point [tex]\((2, 6)\)[/tex] is
[tex]\[
y = 6.
\][/tex]
