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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

3 7 - 4 0 Rotating a Conic A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle θ , write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen.

r = 1 4 3 cos θ ; θ = π 3

To determine

a)

To find:

The eccentricity and the directrix of the conic.

Explanation

Given:

The equation is given as,

r=143cosθ

Approach:

Two formulas representing conics with eccentricity e are,

r=ed1±ecosθ…(1)

Or,

r=ed1±esinθ

Compare the given equation of conic with these standard equations to find the eccentricity.

Then,

For the equation r=ed1±ecosθ corresponding directrix is,

x=±d…(2)

For the equation r=ed1±esinθ corresponding directrix is,

y=±d

Calculation:

Consider the following

To determine

b)

To find:

Resulting equation of the conic obtained after rotation by θ=π3

To determine

c)

To sketch:

The graph of the original conic and rotated conic on the same screen.

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