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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Sketching and Identifying a Conic In Exercises 13–22, find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results.

r = 1 1 + sin θ

To determine

To calculate: The eccentricity and the distance from the pole to the directrix of the conic, r=11+sinθ. Then sketch and identify the graph. By the use of graphing utility verify the results.

Explanation

Given:

The polar equation of conic, r=11+sinθ.

Formula used:

The conic is an ellipse for 0<e<1.

The conic is a parabola for e=1.

The conic is a hyperbola for e>1.

Calculation:

The polar equation of conic is written as,

r=11+sinθ

Compare the above equation with the general equation of conic, r=ed1+esinθ where, e is the eccentricity and |d| is the distance between the focus at the pole and its corresponding directrix. Here, ed=1 and e=1.

Now, find the value of d,

ed=1(1)d=1d=1

Thus, the distance between the focus at the pole and its corresponding directrix is |d|=1.

Graph:

From the above calculation, the eccentricity of conic is e=1. The eccentricity of the conic is equal to one, e=1 then its represent parabola. That means, the conic represents a parabola.

The distance between the focus at the pole and its corresponding directrix is 1. That means, the directrix of a parabola is y=1.

So, sketch the graph of a parabola by plotting points from θ=0 to θ=2π

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