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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

25-28 Polar Equation for a Hyperbola A polar equation of a conic is given.(a) Show that the conic is a hyperbola, and sketch its graph.(b) Find the vertices and directrix, and indicate them on the graph.(c) Find the center of the hyperbola, and sketch the asymptotes.

r = 10 1 4 sin θ

To determine

(a)

To show:

The polar equation r=1014sinθ is a hyperbola and draw the graph of the equation.

Explanation

Approach:

The general equation of conic in polar form with one focus at origin and eccentricity e is given by as below.

r=ed1±ecosθ or r=ed1±esinθ

The conditions for different conic section for above formula follows as below.

a) It is a parabola if e=1.

b) It is an ellipse if 0<e<1.

c) It is hyperbola if e>1.

Calculation:

Consider the given equation,

r=1014sinθ

Compare above equation with general equation of a conic for eccentricity e.

e(=4)>1

Since, eccentricity e is greater than 1, the given conic is a hyperbola.

To sketch the graph, plot the points for θ=0,π,5π4,3π2,7π4. Obtain the values of r for these values of θ.

The following table is obtained.

To determine

(b)

To find:

The vertices and directrix of hyperbola r=1014sinθ and show them on the graph.

To determine

(c)

To find:

The center and asymptotes of hyperbola r=1014sinθ and show on the graph.

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