Read this section and work through practice problems 1-9.
Every line has the property that the slope of the segment between any two points on the line is the same, and this constant slope property of straight lines leads to ways of finding equations to represent nonvertical lines.
In calculus, we will usually know a point on the line and the slope of the line so the point–slope form will be the easiest to apply, and the other forms of equations for straight lines can be derived from the point–slope form.
If is a
nonvertical line through a known point with a known slope m (Fig. 10), then the equation of the line is
Point-Slope:
Example 7: Find the equation of the line through (2,–3) with slope 5.
Solution: The solution is simply a matter of knowing and
using the point–slope formula. and so . This equation simplifies to (Fig. 11).
The equation of a vertical line through a point is . The only points on the vertical line through the point have the same x–coordinate as
.
If two points and are on the line , then we can calculate the slope between them and use the first point and the point–slope equation to get the equation of :
Two Points: where
Once we have the slope , it does not matter whether we use or as the point. Either choice will give the same simplified equation
for the line.
It is common practice to rewrite the equation of the line in the form , the slope-intercept form of the line. The line has slope and crosses the y-axis at
the point ( 0, b ).
Practice 7: Use the definition of slope to calculate the slope of the line .
The point-slope and the two-point formulas are usually more useful for finding the equation of a line, but the slope-intercept form is usually the most useful form for an answer because it allows us to easily picture the graph of the line and to quickly
calculate y-values.
The angle of inclination of a line with the x-axis is the smallest angle θ which the line makes with the positive x-axis as measured from the x-axis counterclockwise to the line (Fig. 12). Since the slope
and since opposite/adjacent, we have that .
The slope of the line is the tangent of the angle of inclination of the line.
Two parallel lines and make equal angles with the x-axis so their angles of inclination will be equal (Fig. 13) and so will their slopes. Similarly, if their slopes and are equal, then the equations of the lines will always differ by a constant:
which is a constant so the lines will be parallel. These two ideas can be combined into a single statement:
Two nonvertical lines and with slopes and are parallel if and only if .
Practice 8: Find the equation of the line in Fig. 14 which contains the point (–2,3) and is parallel to the line .
If two lines are perpendicular and neither line is vertical, the situation is a bit more complicated (Fig. 15).
Assume and are two nonvertical lines that intersect at the origin (for simplicity) and that and are points away from the origin on and , respectively. Then the slopes of and will be and . The line connecting and forms the third side of the triangle
, and this will be a right triangle if and only if and are perpendicular. In particular, and are perpendicular if and only if the triangle satisfies
the Pythagorean theorem:
or
.
By squaring and simplifying, this last equation reduces to
.
We have just proved the following result:
Two nonvertical lines and with slopes and are perpendicular if and only if their slopes are negative reciprocals of each other:
Practice 9: Find the line which goes through the point (2,–5) and is perpendicular to the line .
Example 8: Find the distance (the shortest distance) from the point (1,8) to the
line : .
Solution: This is a sophisticated problem which requires several steps to solve.
First we need a picture of the problem (Fig. 16). We will find the line through the point (1,8) and perpendicular to . Then we will find the point where and intersect, and, finally, we will find the distance from
to (1,8).
(i) has slope 1/3 so has slope , and has the equation which simplifies to .
(ii) We can find the point of intersection of and by replacing the in the equation for with the from so . Then so
, so and intersect at .
(iii) Finally, the distance from to (1,8) is just the distance from the point (1,8) to the point which is .
If two lines which are not perpendicular intersect at a point and neither line is vertical, then we can use some geometry and trigonometry to determine the angles formed by the intersection of the lines (Fig. 17). Since θ2
is an exterior angle of the triangle ABC, θ2 is equal to the sum of the two opposite interior angles so and . Then, from trigonometry,
The inverse tangent of an angle is between and ( –90o and 90o) so always gives the smaller of the angles.
The larger angle is or 180o – θo.
The smaller angle θ formed by two nonperpendicular lines with slopes and is
.
Example 9: Find the point of intersection and the angle between and . (Fig. 18)
Solution: Solving the first equation for y and then substituting into the second equation, so . Putting this back into either equation, we get . Each of the lines is
in the slope–intercept form so it is easy to see that and . Then