   Chapter 10.6, Problem 16E

Chapter
Section
Textbook Problem

Identifying and Sketching a Conic In Exercises 13-22, find the eccentricity and the distance from the pole to the directrix of the conic. Then identify the conic and sketch its graph. Use a graphing utility to confirm your results. r = 4 1 + cos θ

To determine

To Calculate:

The eccentricity and distance from the pole to the directrix of the conic given by the equation r=41+cosθ and identify the conic. Also, sketch its graph and confirm the results with that of the graphing utility

Explanation

Given: The polar equation is given as r=41+cosθ

Formula Used:

For the equation of the type r=ed1+esinθ, e is the eccentricity and d is the distance between focus at pole and corresponding directrix.

Calculation:

The given equation is: r=41+cosθ

And the equation can be re-written as,

r=4×11+1cosθ

Compare this equation with the standard equation r=ed1+ecosθ to get the eccentricity as e=1 and the distance as d=4.

Here e=1, hence the curve for the given equation is parabola.

Hence, the distance of directrix from the pole is d=4

And the equation of directrix is x=4.

Graph:

Since e=1, hence the curve for the given equation is parabola

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