What are the necessary criteria for a line to be perpendicular to the given line and have the same $y$ intercept?
A. The slope is $-\frac{3}{2}$ and contains the point $(0,2)$.
B. The slope is $-\frac{2}{3}$ and contains the point $(0,-2)$.
C. The slope is $\frac{3}{2}$ and contains the point $(0,2)$.
D. The slope is $-\frac{3}{2}$ and contains the point $(0,-2)$.
Answer (2)
A line is perpendicular if its slope is the negative reciprocal of the original line's slope.
The y -intercepts must be equal for the lines to have the same y -intercept.
Check each line to see if any other line is perpendicular and has the same y -intercept.
Conclude that a line must have a negative reciprocal slope and the same y -intercept to satisfy the conditions: negative reciprocal slope and same y -intercept .
Explanation
Problem Analysis Let's analyze the given lines and determine the criteria for a line to be perpendicular to them while sharing the same y -intercept. We have four lines defined by their slopes and y -intercepts (since they contain the point ( 0 , y ) ).
Criteria for Perpendicularity and Same y-intercept A line is perpendicular to another line if its slope is the negative reciprocal of the other line's slope. If the original line has a slope m , the perpendicular line has a slope − m 1 . Also, for the lines to have the same y -intercept, their y -intercept values must be equal.
Checking Each Line Let's examine each of the given lines:
Line 1: Slope m 1 = − 2 3 , y -intercept b 1 = 2 .
The slope of a line perpendicular to Line 1 is m 1 ⊥ = − m 1 1 = − − 2 3 1 = 3 2 .
We look for a line with slope 3 2 and y -intercept 2. None of the given lines match these criteria.
Line 2: Slope m 2 = − 3 2 , y -intercept b 2 = − 2 .
The slope of a line perpendicular to Line 2 is m 2 ⊥ = − m 2 1 = − − 3 2 1 = 2 3 .
We look for a line with slope 2 3 and y -intercept -2. None of the given lines match these criteria.
Line 3: Slope m 3 = 2 3 , y -intercept b 3 = 2 .
The slope of a line perpendicular to Line 3 is m 3 ⊥ = − m 3 1 = − 2 3 1 = − 3 2 .
We look for a line with slope − 3 2 and y -intercept 2. None of the given lines match these criteria.
Line 4: Slope m 4 = − 2 3 , y -intercept b 4 = − 2 .
The slope of a line perpendicular to Line 4 is m 4 ⊥ = − m 4 1 = − − 2 3 1 = 3 2 .
We look for a line with slope 3 2 and y -intercept -2. None of the given lines match these criteria.
Checking for Perpendicularity and Same y-intercept Among Given Lines Now let's check if any of the given lines are perpendicular to each other and have the same y-intercept.
Line 1 and Line 3 have the same y-intercept of 2. However, their slopes are − 2 3 and 2 3 , respectively. These are not negative reciprocals of each other, so they are not perpendicular.
Line 1 and Line 4 have slopes of − 2 3 , so they are parallel, not perpendicular. They also have different y-intercepts.
Line 2 and Line 3: The slope of Line 2 is − 3 2 , and the slope of Line 3 is 2 3 . These are negative reciprocals of each other, so the lines are perpendicular. However, their y-intercepts are -2 and 2, respectively, so they do not have the same y-intercept.
Line 2 and Line 4: The y-intercept of Line 2 and Line 4 is -2. The slope of Line 2 is − 3 2 . The slope of Line 4 is − 2 3 . These are not negative reciprocals of each other, so they are not perpendicular.
Conclusion From the above analysis, we can conclude that for a line to be perpendicular to a given line and have the same y -intercept, it must satisfy two conditions:
Its slope must be the negative reciprocal of the given line's slope.
Its y -intercept must be equal to the given line's y -intercept.
Among the given lines, Line 2 and Line 3 are perpendicular but do not have the same y-intercept. No other pair of lines satisfies both conditions.
Examples
Understanding perpendicular lines and y-intercepts is crucial in various real-world applications. For example, in architecture, ensuring walls are perpendicular to the floor is essential for structural stability. Similarly, in navigation, understanding the y-intercept and slope of a path helps determine the starting point and direction of travel. These concepts are also fundamental in computer graphics, where lines and shapes are defined using slopes and intercepts to create visual representations.
To have a line that is perpendicular to a given line and shares the same y -intercept, the slope must be the negative reciprocal of the given line's slope, and both lines must have the same y -intercept value. Upon analyzing the provided options, none meet these criteria. Therefore, there is no correct answer among the options provided.
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