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Chapter 1: Problem 78

Write equations of the lines through the given point (a) parallel to and (b) perpendicular to the given line. $$x-4=0, \quad(3,-2)$$

Short Answer

Expert verified
The line parallel to \( x=4 \) and passing through the point (3,-2) is \( x=3 \). The line perpendicular to \( x=4 \) and passing through the point (3,-2) is \( y=-2 \).

Step by step solution

01

Determine the line's equation format

First, rewrite the given line's equation in the form \( y = mx + c \), where \( m \) is the slope, \( c \) is the y-intercept. The equation \( x-4 = 0 \) becomes \( x = 4 \)
02

Identify the slope

Since the line's equation is \( x = 4 \), this is a vertical line which means its slope is undefined.
03

Find the parallel line's equation

A parallel line will have the same slope, but pass through the new point. As the line is vertical, another vertical line passing through the point (3,-2) will have the equation \( x=3 \)
04

Find the perpendicular line's equation

For a line that is perpendicular to a vertical line, its slope will be zer0. Find the equation of the line with zero slope passing through the point (3,-2). Using the slope point form \( y - y_1 = m(x - x_1) \), where \( m \) is slope and ( \( x_1, y_1 \) ) is a point on the line, substitute m = 0 and ( \(x_1, y_1\)) = (3,-2). Hence the equation will be \( y = -2 \)

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