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

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

Sagot :

GinnyAnswer
To find the equation of the line that is perpendicular to a given line and passes through a specific point, follow these steps:

1. Identify the Equation of the Given Line:
- The given lines are [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex]. These are vertical lines because their equations are in the form [tex]\( x = \text{constant} \)[/tex].

2. Determine the Slope of the Perpendicular Line:
- A line that is perpendicular to a vertical line must be a horizontal line. This is because vertical lines have undefined slopes, and the slopes of perpendicular lines are negative reciprocals of each other.

3. Understand the Characteristics of Horizontal Lines:
- Horizontal lines have equations of the form [tex]\( y = \text{constant} \)[/tex].

4. Identify the Given Point:
- The given point is [tex]\( (2, 6) \)[/tex]. This point must lie on the perpendicular line.

5. Find the Equation of the Perpendicular Line:
- Since the line must be horizontal and pass through the point [tex]\( (2, 6) \)[/tex], the y-coordinate of all points on this line must be 6. Therefore, the equation of the line is [tex]\( y = 6 \)[/tex].

So, the equation of the line that is perpendicular to the given vertical line [tex]\( x = 2 \)[/tex] and [tex]\( x = 6 \)[/tex] and passes through the point [tex]\( (2, 6) \)[/tex] is
[tex]\[ y = 6 \][/tex]
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