Sure, let's analyze the given options to find which line is perpendicular to another line and passes through the point [tex]\((2, 6)\)[/tex].
1. x = 2: This line is a vertical line that goes through the x-coordinate of 2. It spans from negative infinity to positive infinity along the y-axis. Since it's vertical, it’s not a horizontal line and, therefore, not perpendicular to another horizontal line.
2. x = 6: Similar to [tex]\( x = 2 \)[/tex], this line is also vertical but goes through the x-coordinate of 6. Vertically aligned lines are not perpendicular to each other.
3. y = 2: This line is a horizontal line that goes through the y-coordinate of 2. Since it's horizontal, it’s not perpendicular to another horizontal line.
4. y = 6: This line is a horizontal line that goes through the y-coordinate of 6.
Now, recall that for two lines to be perpendicular, one must be vertical and the other horizontal. Considering the point [tex]\( (2, 6) \)[/tex]:
- If we have the point [tex]\( (2, 6) \)[/tex], the line we are searching for must pass through this point.
- Out of the given choices, the line [tex]\( y = 6 \)[/tex] passes through the y-coordinate of 6, and thus through the point [tex]\( (2, 6) \)[/tex].
- Additionally, a line that is horizontal ([tex]\( y = 6 \)[/tex]) is indeed perpendicular to vertical lines (like [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex]).
So, the equation of the line that is perpendicular to a vertical line, and which passes through the point [tex]\((2, 6)\)[/tex], is [tex]\( y = 6 \)[/tex].
Hence, the correct answer is:
[tex]\[ y = 6 \][/tex]
