1. If a = 6 and b = 4 , find parametric equations for the curve that consists of
ID: 2842939 • Letter: 1
Question
1. If a = 6 and b = 4, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle ? as the parameter.
2. Consider the parametric equations below.
x = ?t
y = 9 - t
(a) Eliminate the parameter to find a Cartesian equation of the curve.
y =
3.Consider the parametric equations below.
x = t2 - 4
y = 1 - 3t
-3 ? t ? 4
(a) Eliminate the parameter to find a Cartesian equation of the curve.
x = ??
??? y ? ???
4.Consider the following.
x = et - 4
y = e2t
(a) Eliminate the parameter to find a Cartesian equation of the curve.
y = ____
x> ___??
5.Consider the following.
x = sin(t)
y = csc(t)
0 < t < ?/2
(a) Eliminate the parameter to find a Cartesian equation of the curve.
y = ______?
y>______?
6.Consider the parametric equations below.
Consider the parametric equations below.
7.Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the parameter.
x = tan(?)
y = sec(?)
(1 , ?2)
y = _____?____
8.Consider the following.
Consider the following.
9.Consider the parametric equations below.
Consider the parametric equations below.
10.Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. (Do this on paper. Your instructor may ask you to turn in this work.)
x = 6sin(t)
y = t2 + t
(0, 0)
y = ???
11.Find dy/dx=??
x = t sin(t)
y = t2 + 5t
12.Consider the following:
Consider the following:
dt If a = 6 and b = 4, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle ? as the parameter. y=8/3 t3/2 dy/dx= d2y/dx2= dy/dx d2y/dx2 infinity infinity Eliminate the parameter. Consider the parametric equations below. Eliminate the parameter to find a Cartesian equation of the curve. Consider the parametric equations below.
Explanation / Answer
Let O denote the origin.
P's x coordinate is the same as B's x coordinate, correct?
Consider the right triangle OAB (we know it's a right triangle because AB is tangent to the circle and OA is a radius, so angle OAB is a right angle).
OA = a
cos(?) = OA/OB = a/OB
So OB = a/cos(?) = a sec(?)
This correctly gives P's x-coordinate even if sec(?) < 0 (so that we can no longer talk of lengths of triangle sides)
Let C denote the point where OA intersects the smaller circle. Drop a perpendicular line segment from C down to the x-axis, meeting it at point D. Note that CD = PB.
Consider triangle OCD. We have
sin(?) = CD/b, so CD = b sin(?)
So P's y-coordinate is b sin(?)
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