Question

In Exercises \(31-34,\) write an equation for the line through \(P\) that is (a) parallel to \(L,\) and (b) perpendicular to \(L .\) $$P(-2,4), \quad L : x=5$$

Short answer

The equation of the line that is parallel to \(L\) and passes through \(P\) is \(x = -2\). The equation of the line that is perpendicular to \(L\) and passes through \(P\) is \(y = 4\).

Step-by-step solution

Identify the Line

Line \(L\) is given by the equation \(x = 5\), so it's a vertical line passing through (5,0) on the x-axis.

Line Parallel to \(L\)

Since parallel lines have the same slope, and the slope of line \(L\) is undefined, the line parallel to \(L\) through point \(P(-2,4)\) will be another vertical line. The equation of a vertical line is of the form \(x = c\), where \(c\) is the x-coordinate of any point on the line. Here, we use the x-coordinate of point \(P\), so the equation for a line parallel to \(L\) passing through \(P\) is \(x = -2\).

Line Perpendicular to \(L\)

Perpendicular lines have opposite reciprocal slopes. For this case, since \(L\) is vertical, the line perpendicular to it would be a horizontal line. The equation of a horizontal line is of the form \(y = c\), where \(c\) is the y-coordinate of any point on the line. Here, we use the y-coordinate of point \(P\), so the equation for a line perpendicular to \(L\) passing through \(P(-2,4)\) is \(y = 4\).

Key concepts

Parallel and Perpendicular Lines

Understanding parallel and perpendicular lines is crucial when dealing with geometric figures and their equations. By definition, two lines are parallel if they never intersect, no matter how far they are extended. This implies that they have the same slope.

On the other hand, two lines are perpendicular to each other if they intersect at a right angle (90 degrees). To be perpendicular, the slopes of the two lines must be negative reciprocals of each other; if the slope of one line is 'm', then the slope of the line perpendicular to it will be '-1/m'.

For example, in our exercise with line L: x=5, because it's a vertical line, any line that is parallel to it will also be a vertical line and hence, does not have a defined slope. Conversely, any line that is perpendicular to it will be a horizontal line, which has a slope of zero.

Equation of a Vertical Line

A vertical line is unique in coordinate geometry for its characteristic of having an undefined slope and it cannot be described by the typical y = mx + b equation. Instead, the equation of a vertical line is simply x = c, where 'c' represents the constant x-coordinate for all points on that line. This form is very straightforward and effective for identifying vertical lines.

In our exercise, the line L: x=5 is vertical. Hence, a line parallel to L through a point P(-2, 4) would also be vertical, represented by the equation x = -2, which signifies all points having an x-coordinate of -2.

Equation of a Horizontal Line

Conversely to vertical lines, a horizontal line demonstrates a slope of zero and can be expressed by an equation in the form of y = c. Here, 'c' is the constant y-coordinate of all points lying on the line. Such an equation signifies that y-value remains consistent throughout the horizontal stretch of the line, making its graph a straight line parallel to the x-axis.

From the exercise, the line perpendicular to L: x=5 is horizontal and passes through the point P(-2, 4). Therefore, the equation of this perpendicular line is y = 4, indicating a horizontal line that crosses the y-axis at 4.

Slope of a Line

The slope of a line is a measure of its steepness and can be calculated using the formula (change in y)/(change in x), often denoted as 'm'. Slope is crucial because it determines the direction and angle at which a line tilts. If a line moves upwards to the right, it has a positive slope; if it moves downwards to the right, the slope is negative. A horizontal line has a slope of zero because there is no change in y-value as x increases, whereas a vertical line has an undefined slope because the change in x-value is zero.

As seen previously, for vertical and horizontal lines, the concept of slope applies differently. The vertical line L: x=5 has an undefined slope, while any horizontal line, like the one found in the exercise that passes through P(-2, 4) and is perpendicular to L, has a slope of zero.