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OTHER PARAMETRIC EQUATIONS EXAMPLE.
Find the equations of the tangent line and
the normal line and the lengths of the tangent and the normal for the curve
represented by
and
at
given that
and
SOLUTION. Since t equals 1 we write x=1 and y=3 and
so that
The equation of the tangent line when t is equal to 1 is y3=1(x1) y=x+2 The equation of the normal line is y 3 = 1(x 1) y= x+4 The
length of the tangent is
The
length of the normal is
EXAMPLE: Find the equations of the tangent line and the normal line and the
lengths of the tangent and the normal to the curve represented by the
parametric equations
and
at
the point where
given
that
and
SOLUTION.
We know that
Then
at the point where
, we have
If
is
substituted in the parametric equations, then
and
The equation of the tangent line when
is
or
The equation of the normal is
or xy=0 The length of the tangent is
The length of the normal is
The horizontal and vertical tangents of a curve can be
found very easily when the curve is represented by parametric equations. The
slope of a curve at any point equals zero when the tangent line is parallel to
the X axis. In parametric equations, if x = x(t) and y = y(t), then the
horizontal and vertical tangents can be found easily by setting
and ^{
} For the horizontal tangent solve
equals zero for t and for the
vertical tangent solve
equals zero for t. EXAMPLE:
Find the points of contact of the horizontal
and the vertical tangents to the curve represented by the parametric equations
and
Plot the graph of the curve by taking
from 0° to 360° in increments of 30°, given
that
and
Figure 37.Ellipse. SOLUTION: The graph of the curve shows that the figure is
an ellipse, figure 37; consequently, it will have two horizontal and two
vertical tangents. The coordinates of the horizontal tangent points are found
by first setting
This gives
so that
and
Substituting 0°, we have
and
Substituting 180°, we obtain
and
The coordinates of the points of contact of the horizontal
tangents to the ellipse are (3,1) and (3,7). The coordinates of the vertical tangent points of contact
are found by setting
We find
from which
Substituting 90°, we obtain
and
Substituting 270° gives
and
The coordinates of the points of contact of the vertical
tangents to the ellipse are ( 1,4) and (7,4). 