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C-86 Appendix C
Finally we mention another pair of parametric equations for the unit circle,
that is, the circle in the x-y plane centered at the origin with radius 1.
Consider the line passing through the point (1, 0) on the unit circle with slope
m. In Exercise 6 you will show that this line intersects the unit circle at the point
P a 1 m2
See Figure E. This construction is equivalent to the one
1 m , 2m
2
discussed in the Writing Mathematics section at the end of Chapter 10, and can
also be used to generate all Pythagorean triples. From this construction we
obtain our third pair of parametric equations for the unit circle:
1 m 2 b . y
x a 1 m2
1 m b 2
d
for q m q
2m
y a
1 m b 2
(6)
Slope m
1-m@ 2m
P ”
1+m@ , ’
1+m@
(_1, 0)
x
≈+¥=1
Figure E
Exercises
1. Let A 8a, b9 Z 0 be a direction vector of a line through the point (x 0 , y 0 ). If
a 0 then what kind of line is it? What is its equation? What about b 0?
2. Consider the line in the x-y plane passing through the point (2, 3) with a
direction vector 84, 29.
(a) Find a vector equation for the line. Sketch the line with Cartesian and
vector notation as in Figure B.
(b) Find parametric equations for the line. Find the x- and y-intercepts of
the line.
(c) Find, if possible, the symmetric equations for the line.
3. In equations (5) why does s vary from 0 to 2pa?
4. Consider the circle in the x-y plane centered at the origin with radius 12.
(a) Use equations (4) to find parametric equations for this circle.
(b) Find an arc-length parameterization for this circle.
5. Consider the circle in the x-y plane centered at the point (h, k) with radius a.
(a) Use a fixed distance and a variable angle to find parametric equations
for this circle. Hint: Modify Figure C and equations (4) by “translation.”
Sketch this circle and label it as in Figure C.
(b) Find an arc-length parameterization for this circle.