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 (2)
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 in point-slope form is: y + 3 = 4 1 ( x + 4 )
Explanation
Understanding the Problem We are given a line in point-slope form and a point, and we need to find the equation of a line perpendicular to the given line that passes through the given point.
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 of 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 a line with slope m 1 is m 2 = − m 1 1 . Therefore, the slope of the line perpendicular to the given line is m 2 = − − 4 1 = 4 1 .
Writing the Equation of the Perpendicular Line Now we need to write the equation of the line with slope 4 1 that passes through the point ( − 4 , − 3 ) . We use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point and m is the slope. Plugging in the values, we get y − ( − 3 ) = 4 1 ( x − ( − 4 )) , which simplifies to y + 3 = 4 1 ( x + 4 ) .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. This problem demonstrates how to find the equation of a line perpendicular to another, which can be applied to ensure structural integrity in real-world constructions.
To find the equation of the line perpendicular to the given line that passes through ( − 4 , − 3 ) , we first determine the slope of the given line as − 4 . The slope of the perpendicular line is then calculated as 4 1 , leading to the final equation y + 3 = 4 1 ( x + 4 ) , which corresponds to option C.
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