   Chapter 10.6, Problem 15E

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 = 7 4 + 8 sin θ

To determine

To Calculate:

The eccentricity and distance from the pole to the directrix of the conic r=74+8sinθ 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=74+8sinθ.

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=74+8sinθ

And the equation can be re-written as,

r=78×21+2sinθ

Compare this equation with the standard equation r=ed1+esinθ to get the Eccentricity as e=2 and the distance as d=78.

Here e>1, hence the curve for the given equation is hyperbola.

Hence, the distance of directrix from the pole is d=78.

Graph:

The given equation is of hyperbola since e>1

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