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Unformatted text preview: MATH 153
Selected Solutions for §10.1 Exercise 12: Eliminate the parameter to ﬁnd a Cartesian equation of the
curve, then sketch the curve and indicate with an arrow the direction in which
the curve is traced as the parameter increases.
x = 4 cos θ, y = 5 sin θ, −π/2 ≤ θ ≤ π/2 Solution: As this parametrization is similar to that of a circle, we use the
same trigonometric identity as before:
sin2 θ + cos2 θ = 1 ⇔
x2 y 2
You may recall that this is the equation of an ellipse, with axes of lengths 4
and 5. As θ only traces out half a period of sine and cosine, we’ll only get half
of the ellipse: 4 2 1 2 3 4 2 4 Figure 1. A sketch of x = 4 cos θ, y = 5 sin θ.
1 2 Exercise 42: If a and b are ﬁxed numbers, ﬁnd parametric equations for the
curve that consists of all possible positions of the point P in the ﬁgure (see
book), using the angle θ as the parameter. The line segment AB is tangent
to the larger circle.
Solution: It should be fairly obvious, after dropping an altitude for the radius
at angle θ of the inner circle, that the y -coordinate of the point is just
y = b sin θ.
Looking at the triangle OAB , which is a right triangle with right angle A, we
see that cos θ = a/x, or that
x = a sec θ.
Thus, a combined parametrization for the possible points P would be
x = a sec θ, y = b sin θ, 0 ≤ θ ≤ 2π. 2 1 4 2 2 4 1 2 Figure 2. A sketch for a = 2, b = 1 (the black curve). ...
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