Explanation Given: The equation of ellipse is: x 2 a 2 + y 2 b 2 = 1 And the following figure is given: Proof: Consider the equation of ellipse and the figure provided: x 2 a 2 + y 2 b 2 = 1 … ( 1 ) From the figure, we see that y x = tan θ y = x tan θ … ( 2 ) Now, put this value of y in equation (1), we get x 2 a 2 + x 2 tan 2 θ b 2 = 1 x 2 a 2 + x 2 sin 2 θ b 2 c o s 2 θ = 1 ( a 2 sin 2 θ + b 2 cos 2 θ ) x 2 = a 2 b 2 cos 2

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 12.1, Problem 109E

To determine

To prove:

The parameterization of the ellipse by the angle $θ$ is:

$x = \frac{a b \cos θ}{\sqrt{a^{2} \sin^{2} θ + b^{2} \cos^{2} θ}}$

$y = \frac{a b s i n θ}{\sqrt{a^{2} \sin^{2} θ + b^{2} \cos^{2} θ}}$

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Chapter 12.1, Problem 108E

Chapter 12.2, Problem 1PQ

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Chapter 12 Solutions

CALCULUS (CLOTH)

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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To prove: The parameterization of the ellipse by the angle θ is: x = a b cos θ a 2 sin 2 θ + b 2 cos 2 θ y = a b s i n θ a 2 sin 2 θ + b 2 cos 2 θ

Solution Summary: The author explains how the parameterization of the ellipse by the angle is proved.

Videos

To prove:

The parameterization of the ellipse by the angle $θ$ is:

$x = \frac{a b \cos θ}{\sqrt{a^{2} \sin^{2} θ + b^{2} \cos^{2} θ}}$

$y = \frac{a b s i n θ}{\sqrt{a^{2} \sin^{2} θ + b^{2} \cos^{2} θ}}$

Want to see the full answer?

Chapter 12 Solutions

To prove: The parameterization of the ellipse by the angle θ is: x = a b cos θ a 2 sin 2 θ + b 2 cos 2 θ y = a b s i n θ a 2 sin 2 θ + b 2 cos 2 θ

Solution Summary: The author explains how the parameterization of the ellipse by the angle is proved.

Videos

To prove: The parameterization of the ellipse by the angle θ is: x=abcosθa2sin2θ+b2cos2θ y=absinθa2sin2θ+b2cos2θ

Want to see the full answer?

Chapter 12 Solutions

To prove:

The parameterization of the ellipse by the angle $θ$ is:

$x = \frac{a b \cos θ}{\sqrt{a^{2} \sin^{2} θ + b^{2} \cos^{2} θ}}$

$y = \frac{a b s i n θ}{\sqrt{a^{2} \sin^{2} θ + b^{2} \cos^{2} θ}}$