Which of the following is the equation of the line that is perpendicular to [tex]y=-\frac{1}{4} x-2[/tex] and goes through the point [tex](-2,-3)[/tex]?
A. [tex]y=-4 x-11[/tex]
B. [tex]y=4 x+5[/tex]
C. [tex]y=-\frac{1}{4} x-\frac{7}{2}[/tex]
D. [tex]y=\frac{1}{4} x-\frac{5}{2}[/tex]
Answer (1)
Determine the slope of the given line: m 1 = − 4 1 .
Calculate the slope of the perpendicular line: m 2 = − m 1 1 = 4 .
Use the point-slope form with the point ( − 2 , − 3 ) and slope 4 : y − ( − 3 ) = 4 ( x − ( − 2 )) .
Simplify to slope-intercept form: y = 4 x + 5 . The final answer is y = 4 x + 5 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to a given line and passes through a specific point. We'll use the properties of perpendicular lines and the point-slope form to determine the equation.
Finding the Slope of the Perpendicular Line The given line is y = − 4 1 x − 2 . The slope of this line is − 4 1 . A line perpendicular to this line will have a slope that is the negative reciprocal of − 4 1 , which is 4 .
Using the Point-Slope Form We are given the point ( − 2 , − 3 ) that the perpendicular line passes through. We can use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the point.
Substituting Values Substituting the slope m = 4 and the point ( − 2 , − 3 ) into the point-slope form, we get:
y − ( − 3 ) = 4 ( x − ( − 2 ))
y + 3 = 4 ( x + 2 )
Simplifying the Equation Now, we simplify the equation to slope-intercept form ( y = m x + b ):
y + 3 = 4 x + 8
y = 4 x + 8 − 3
y = 4 x + 5
Final Answer The equation of the line that is perpendicular to y = − 4 1 x − 2 and passes through the point ( − 2 , − 3 ) is y = 4 x + 5 .
Examples
Imagine you're designing a rectangular garden and need to ensure the paths are perfectly perpendicular to the edges. Knowing the slope of one edge, you can calculate the slope of the path to ensure it's at a right angle. This principle applies in various fields, from architecture to navigation, where maintaining perpendicularity is crucial for accuracy and stability. Understanding perpendicular lines helps in creating precise and balanced designs in real-world applications.
