What is the equation of the line that is perpendicular to the given line and passes through the point $(3,4)$?
$y=-\frac{1}{3} x+5$
$y=\frac{1}{3} x+3$
$y=3 x+2$
$y=3 x-5
Answer (2)
The slope of the given line y = − 3 1 x + 5 is − 3 1 .
The slope of the perpendicular line is the negative reciprocal of − 3 1 , which is 3 .
Using the point-slope form with point ( 3 , 4 ) and slope 3 , the equation is y − 4 = 3 ( x − 3 ) .
Converting to slope-intercept form, the equation of the perpendicular line is y = 3 x − 5 .
Explanation
Analyze the problem The given line is y = − 3 1 x + 5 . We want to find the equation of a line that is perpendicular to this line and passes through the point ( 3 , 4 ) .
Find the slope of the perpendicular line The slope of the given line is − 3 1 . The slope of a line perpendicular to this line is the negative reciprocal of − 3 1 , which is 3 .
Use the point-slope form Now we 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. In our case, m = 3 and ( x 1 , y 1 ) = ( 3 , 4 ) . So the equation is y − 4 = 3 ( x − 3 ) .
Convert to slope-intercept form Now we convert the equation to slope-intercept form, which is y = m x + b . We have y − 4 = 3 ( x − 3 ) , so y − 4 = 3 x − 9 . Adding 4 to both sides, we get y = 3 x − 9 + 4 , which simplifies to y = 3 x − 5 .
State the final answer Therefore, the equation of the line that is perpendicular to the given line and passes through the point ( 3 , 4 ) is y = 3 x − 5 .
Examples
Understanding perpendicular lines is crucial in various real-world applications. For example, architects use this concept to ensure walls are built at right angles for structural stability. Similarly, in navigation, knowing the perpendicular direction to a path helps in calculating the shortest distance to a destination. This principle also applies in computer graphics, where creating orthogonal projections relies on perpendicular lines to render 3D objects accurately on a 2D screen.
The equation of the line that is perpendicular to the given line y = − 3 1 x + 5 and passes through the point ( 3 , 4 ) is y = 3 x − 5 .
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