A given line has the equation [tex]$2 x+12 y=-1$[/tex]. What is the equation, in slope-intercept form, of the line that is perpendicular to the given line and passes through the point [tex]$(0,9)$[/tex]?
A. [tex]$y=-6 x+9$[/tex]
B. [tex]$y=-\frac{1}{6} x+9$[/tex]
C. [tex]$y=\frac{1}{6} x+9$[/tex]
D. [tex]$y=6 x+9$[/tex]
Answer (2)
Convert the given equation to slope-intercept form: y = − 6 1 x − 12 1 .
Determine the slope of the perpendicular line by taking the negative reciprocal of the given line's slope: m = 6 .
Use the point-slope form with the point ( 0 , 9 ) to find the equation of the perpendicular line.
Write the equation in slope-intercept form: y = 6 x + 9 .
Explanation
Understanding the Problem We are given the equation of a line 2 x + 12 y = − 1 and we need to find the equation of a line that is perpendicular to this line and passes through the point ( 0 , 9 ) . The final equation should be in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Finding the Slope of the Given Line First, let's convert the given equation to slope-intercept form to find its slope. We have 2 x + 12 y = − 1 . Solving for y , we get:
12 y = − 2 x − 1
y = 12 − 2 x − 1
y = − 6 1 x − 12 1
Calculating the Slope of the Perpendicular Line The slope of the given line is − 6 1 . The slope of a line perpendicular to this line is the negative reciprocal of this slope. Therefore, the slope of the perpendicular line is:
m = − − 6 1 1 = 6
Finding the Equation of the Perpendicular Line Now we know that the perpendicular line has a slope of 6 and passes through the point ( 0 , 9 ) . Since the x-coordinate of the point is 0 , this point is the y-intercept. Therefore, b = 9 . We can now write the equation of the perpendicular line in slope-intercept form:
y = m x + b
y = 6 x + 9
Examples
Imagine you're designing a rectangular garden and need to ensure the paths intersect at a perfect right angle for aesthetic appeal and efficient space usage. This problem helps you determine the equation of one path given the equation of the other and a specific point it needs to pass through. Understanding perpendicular lines is crucial in various fields like architecture, construction, and even video game design, where precise angles and spatial relationships are essential.
To find the perpendicular line to 2 x + 12 y = − 1 that passes through ( 0 , 9 ) , we found its slope to be 6. The resulting equation of the perpendicular line is y = 6 x + 9 , which corresponds to option D. Thus, the final answer is D.
;
