Let's consider each of the given lines and determine the equations of the lines that are perpendicular to them and pass through the point [tex]\((2, 6)\)[/tex].
1. Line [tex]\( x = 2 \)[/tex]:
The line [tex]\( x = 2 \)[/tex] is a vertical line, which means any line perpendicular to [tex]\( x = 2 \)[/tex] must be a horizontal line. A horizontal line has an equation of the form [tex]\( y = k \)[/tex], where [tex]\( k \)[/tex] is a constant.
Since the line must pass through the point [tex]\((2, 6)\)[/tex], the value of [tex]\( y \)[/tex] for this point is 6. Therefore, the equation of the line perpendicular to [tex]\( x = 2 \)[/tex] and passing through [tex]\((2, 6)\)[/tex] is:
[tex]\[
y = 6
\][/tex]
2. Line [tex]\( x = 6 \)[/tex]:
Similar to the line [tex]\( x = 2 \)[/tex], [tex]\( x = 6 \)[/tex] is also a vertical line. Thus, the equation of the line perpendicular to [tex]\( x = 6 \)[/tex] and passing through [tex]\((2, 6)\)[/tex] must also be a horizontal line. Since it passes through [tex]\((2, 6)\)[/tex], the equation remains:
[tex]\[
y = 6
\][/tex]
3. Line [tex]\( y = 2 \)[/tex]:
The line [tex]\( y = 2 \)[/tex] is a horizontal line, which means any line perpendicular to [tex]\( y = 2 \)[/tex] must be a vertical line. A vertical line has an equation of the form [tex]\( x = k \)[/tex], where [tex]\( k \)[/tex] is a constant.
Since the line must pass through the point [tex]\((2, 6)\)[/tex], the value of [tex]\( x \)[/tex] for this point is 2. Therefore, the equation of the line perpendicular to [tex]\( y = 2 \)[/tex] and passing through [tex]\((2, 6)\)[/tex] is:
[tex]\[
x = 2
\][/tex]
4. Line [tex]\( y = 6 \)[/tex]:
Similar to the line [tex]\( y = 2 \)[/tex], [tex]\( y = 6 \)[/tex] is also a horizontal line. Thus, the equation of the line perpendicular to [tex]\( y = 6 \)[/tex] and passing through [tex]\((2, 6)\)[/tex] must also be a vertical line. Since it passes through [tex]\((2, 6)\)[/tex], the equation remains:
[tex]\[
x = 2
\][/tex]
In summary, the equations of the lines that are perpendicular to the given lines and pass through the point [tex]\((2, 6)\)[/tex] are:
- For lines [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex]: [tex]\( y = 6 \)[/tex]
- For lines [tex]\( y = 2 \)[/tex] and [tex]\( y = 6 \)[/tex]: [tex]\( x = 2 \)[/tex]
