What is the equation of the line that is perpendicular to the given line and passes through the point [tex][tex]$(2,6)$[/tex][/tex]?

A. [tex][tex]$x=2$[/tex][/tex]
B. [tex][tex]$x=6$[/tex][/tex]
C. [tex][tex]$y=2$[/tex][/tex]
D. [tex][tex]$y=6$[/tex][/tex]



Answer :

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]