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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 = 9 2 + 2 cos θ ; θ = 5 π 6

To determine

a)

To find:

The eccentricity and the directrix of the conic r=92+2cosθ.

Explanation

Given:

The equation is given as,

r=92+2cosθ

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.

Based on this, determine whether the given conic is a parabola, ellipse or a hyperbola.

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

x=±d

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

y=±d(2)

Calculation:

Consider the following

To determine

b)

To find:

Resulting equation of the conic obtained after rotation by θ=5π6

To determine

c)

To sketch:

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

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