Question
Find equations of the lines that pass through the given point and are parallel to and perpendicular to the given line. \( x-2=0,(7,-1) \) (a) parallel to the given line (b) perpendicular to the given line (
Ask by Simmons Chandler. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
(a) The line parallel to x = 2 passing through (7, -1) is x = 7.
(b) The line perpendicular to x = 2 passing through (7, -1) is y = -1.
Solution
The given line is x - 2 = 0, which can be rewritten as x = 2. This represents a vertical line where the x-coordinate is always 2.
(a) Lines parallel to a vertical line are also vertical. Therefore, any line parallel to x = 2 will have the form x = k, where k is a constant. Since the line must pass through the point (7, -1), the x-coordinate of the point, 7, must equal k. Thus, the equation of the line is:
x = 7
(b) Lines perpendicular to a vertical line are horizontal lines. Horizontal lines have equations of the form y = c, where c is a constant. Since the line must pass through (7, -1), the y-coordinate of the point, -1, must equal c. Thus, the equation of the line is:
y = -1
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Simplify this solution Beyond the Answer
The line \( x - 2 = 0 \) is a vertical line that has an undefined slope. Since a line parallel to this would also be vertical, the equation of the parallel line through the point \( (7, -1) \) is simply \( x = 7 \). For the perpendicular line, it would be a horizontal line because the slopes of vertical and horizontal lines are negative reciprocals. The slope of the horizontal line is \( 0 \), so the equation of the perpendicular line passing through the point \( (7, -1) \) is \( y = -1 \).
