EQUATIONS AND LENGTHS OF TANGENTS AND NORMALS
In
figure 35, the coordinates of point P_{1} on the curve are (x_{1},y_{1}).
Let the slope of the tangent line to the curve at point P_{1} be
denoted by m_{1}. If you know the slope and a point through which the
tangent line passes, you can determine the equation of that tangent line by
using the pointslope form.
Thus,
the equation of the tangent line, MP_{1},
is
The
normal to a curve at a point (x_{1},y_{1}) is the line
perpendicular to the tangent line at that point. The slope of the normal line,
m_{2}, is
Figure 3S.Curve with tangent and normal lines.
where,
as before, the slope of the tangent line is m_{1}. This is shown in the
following:
If
then
therefore,
The equation of the normal line through
P_{1} is
Notice that since the slope of the tangent line is m, and
the slope of the line which is normal to the tangent is m_{2} and
then the product of the slopes of the tangent and normal
lines equals 1. The relationship between the slopes of the tangent and normal
lines is stated more formally as follows: The slope of the normal line is
the negative reciprocal of the slope of the tangent line.
Another approach to show the relationship between the
slopes of the tangent and normal lines follows:
The inclination of one line must be 90^{0}
greater than the other. Then
if
then
We know from trigonometry that
Therefore,
The length of the tangent is defined
as that portion of the tangent line between the point P_{1} (x_{1},y_{1})
and the point where the tangent line crosses the X axis. In figure 35, the
length of the tangent is the line MP,.
The length of the normal is defined
as that portion of the normal line between the point P_{1} and the X
axis; that is the line, P_{1}R_{1} which is perpendicular to
the tangent line.
As shown in figure 35,.the lengths of the tangent and
normal may be found by using the Pythagorean theorem.
From triangle MP_{1}N_{,} in
figure 35,
and
Since
then
In triangle NP_{1}R,
and
Thus, the length of the tangent is equal to
and the length of the normal is equal to
EXAMPLE. Find the equation of the tangent line, the equation
of the normal line, and the lengths of the tangent and the normal of
SOLUTION: Find the value of 2a from
Since
_{
}
then
The slope is
^{}
Using the pointslope form of a straight line, we have
then, at point (3,2)
and
which is the equation of the tangent line.
Use the negative reciprocal of the slope to find the
equation of the normal line as follows:
then
To find the length of the tangent, we use the Pythagorean
theorem. Thus, the length of the tangent is
The length of the normal is equal to
PRACTICE PROBLEMS:
Find the equations of the tangent line and the normal
line, and the lengths of the tangent and the normal for the following:
