   Chapter 10.6, Problem 21E

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 = 300 − 12 + 6 sin θ

To determine

To Calculate:

The eccentricity and distance from the pole to the directrix of the conic for the equation r=30012+6sinθ 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=30012+6sinθ

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=30012+6sinθ

And the equation can be re-written as,

r=30012(112sinθ)

r=12×50(112sinθ)

Compare this equation with the standard equation r=ed1esinθ to get the eccentricity as e=12 and the distance as |d|=50

Here e=12<1, hence the curve is ellipse

And the distance of directrix from the pole is |d|=50

Graph:

Use Desmos graphing calculator and write the equation, plot the following graph:

To draw the graph of an Ellipse, plot some points by taking different values for θ as in the table

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