1.7 Homework Hints
Given a parametric curve
x = f (t),
y = g(t),
a≤t≤b
we are asked to ”eliminate the parameter to find a Cartesian equation of the curves.” Here, the parameter
is t. This parametric curve is defined by two equations (x = .., y = ..) in three variables (x, y, and t). This
question is asking us to find a single equation in two variables x and y which describes the same curve.
Example. Suppose we have the parametric curve
x = cos(t), y = sin(t), 0 ≤ t ≤ 2π
This is the unit circle example we went over in class. If we were to eliminate t and produce a single equation
in (x, y) which describes this curve, we would come up with
x2 + y 2 = 1
which is the equation of a circle with radius 1. The key point is that the parametric equations ( x = .. and
y = ..) and the region described by the cartesian equation (the equation of a circle) give the same curve (the
unit circle).
How to eliminate t
There are many many tricks for eliminating t. I will go over three which will allow you to do every problem
in your homework.
1. Solve for t in the equation x = f (t). Plug this t value into y and simplify. (Or: solve for t in y = g(t)
and plug into x = f (t)).
2. Use the trigonometric identity cos 2 (ω) + sin2 (ω) = 1.
3. Solve for x in terms of y or y in terms of x
Example. ”Solve for t in the equation x = f (t). Plug this t value into y and simplify. (Or: solve for t in
y = g(t) and plug into x = f (t)).”
x = 1 + 3t, y = 2 − t2
Solving for t in the y equation is hard (since you have to take a square root). So I will solve for t in the x
equation:
x = 1 + 3t implies that t =
Now I will plug this equation for t into y:
y
2 − t2
2
x−1
= 2−
3
2
x−1
= −
+2
3
x−1
3
=
1
plugging in t
So my final answer is
y=−
x−1
3
2
+2
(This is a parabola by the way. If that is not obvious to you, think about how you would modify the equation
y = x2 to get the equation above).
Example. ”Use the trigonometric identity cos 2 (ω) + sin2 (ω) = 1”
x = 3 sin(2t),
y = 3 cos(2t),
0≤t≤
π
2
This example is very similar to the circle of radius 3 (take the unit circle, multiply both of the parametric
equations by 3). However, sin and cos are switched. Now suppose we plug in these parametric equations to
x2 + y 2 :
x2 + y 2
= (3 sin(2t))2 + (3 cos(2t))2
= 9 sin(2t)2 + 9 cos(2t)2
= 9(sin(2t)2 + cos(2t)2 )
= 9(1)
=
So my final answer is
9
x2 + y 2 = 9
This is a circle of radius 3, centered at (0, 0). However, because of the restriction 0 ≤ t ≤
curve might not be the entire circle. Note that:
π
2,
the plot of this
π
implies that 0 ≤ 2t ≤ π
2
(I multiplied everything by 2). So we can plug in the beginning and end points of this curve. At t = 0,
2t = 0 as well, and the beginning point is
0≤t≤
x = 3 sin(0), y = 3 cos(0)
In other words, the point (0, 3). At t =
π
2,
2t = π, and the end point is
x = 3 sin(π), y = 3 cos(π)
In other words, the point (0, −3). So this curve is the part of the circle of radius 3 that starts at (0, 3) and
goes down to (0, −3). Which side of the circle does that include (right or left)? Plug in t = π2 to find out!
Example. ”Solve for x in terms of y or y in terms of x”
1
x = e 2 t,
1
y = et
1
Here, note that x = (et ) 2 = y 2 (make sure you know why!). Hence our final answer is
x=
2
√
y