What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point $(-4,-3)$?
A. $y+3=-4(x+4)$
B. $y+3=-\frac{1}{4}(x+4)$
C. $y+3=\frac{1}{4}(x+4)$
D. $y+3=4(x+4)$
Answer (1)
Determine the slope of the given line: m 1 = − 4 .
Calculate the slope of the perpendicular line: m 2 = 4 1 .
Use the point-slope form with the point ( − 4 , − 3 ) and the perpendicular slope: y + 3 = 4 1 ( x + 4 ) .
The equation of the perpendicular line is y + 3 = 4 1 ( x + 4 ) .
Explanation
Understanding the Problem We are given a line and a point, and we want to find the equation of a line that is perpendicular to the given line and passes through the given point. The equation should be in point-slope form.
Finding the Slope of the Given Line The given line is y + 3 = − 4 ( x + 4 ) . This is in point-slope form, y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. In this case, the slope of the given line is m 1 = − 4 .
Finding the Slope of the Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. So, the slope of the perpendicular line is m 2 = − m 1 1 = − − 4 1 = 4 1 .
Finding the Equation of the Perpendicular Line in Point-Slope Form We are given the point ( − 4 , − 3 ) that the perpendicular line passes through. Using the point-slope form of a line, y − y 1 = m ( x − x 1 ) , we have y − ( − 3 ) = 4 1 ( x − ( − 4 )) , which simplifies to y + 3 = 4 1 ( x + 4 ) .
Final Answer Therefore, the equation of the line in point-slope form that is perpendicular to the given line and passes through the point ( − 4 , − 3 ) is y + 3 = 4 1 ( x + 4 ) .
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and navigation. For example, when designing a building, architects need to ensure that walls are perpendicular to the ground for stability. Similarly, in navigation, understanding perpendicular paths helps in determining the shortest distance between two points or avoiding obstacles.
