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(y − 2 ) 2 + (x − 3 ) 2 = 1
2 2 3 2

This is the equation of an ellipse centered at x = 3, y = 2 and having major and
minor radii equal to 3 and 2, respectively.

Question 1: What are the coordinates of the foci of this ellipse?

Question 2: Compare the above curve with the curve defined through:

x = 3 + 3 cos(2t), y = 2 + 2 sin(2t), and 0 < t < 2π

What conclusions can you draw from your answer?

In-Class Exercises
Pb. 1.3 Show that the following parametric equations:

x = h + a sec(t), y = k + b tan(t), and –π/2 < t < π/2

are those of the hyperbola also represented by the equation:

(xh− ) 2 − (y − ) k 2 = 1
a 2 b 2

Pb. 1.4 Plot the hyperbola represented by the parametric equations of Pb.
1.3, with h = 2, k = 2, a = 1, b = 2. Find the coordinates of the vertices and the
foci. (Hint: One branch of the hyperbola is traced for –π/2 < t < π/2, while the
other branch is traced when π/2 < t < 3π/2.)
Pb. 1.5 The parametric equations of the cycloid are given by:

x = Rωt + R sin(ωt), y = R + R cos(ωt), and 0 < t

Show how this parametric equation can be obtained by following the kine-
matics of a point attached to the outer rim of a wheel that is uniformly rolling,
without slippage, on a flat surface. Relate the above parameters to the linear
speed and the radius of the wheel.
Pb. 1.6 Sketch the curve C defined through the following parametric equa-
tions:

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