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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 = 2 5 3 sin θ ; θ = 2 π 3

To determine

a)

To find:

The eccentricity and the directrix of the conic.

Explanation

Given:

The equation is given by,

r=253sinθ, θ=2π3

Approach:

Two formulas representing conics with eccentricity e are,

r=ed1±ecosθ

or,

r=ed1±esinθ(1)

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

Calculation:

Consider the following.

r=253sinθ

Divide the numerator and denominator by 5

To determine

b)

To find:

Resulting equation of the conic obtained after rotation by θ=2π3.

To determine

c)

To sketch:

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

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