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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Finding a Polar Equation In Exercises 39-44. Find a polar equation for the conic with its focus at the pole and the given vertex or vertices.

Conic Vertex or Vertices

Ellipse ( 2 , π 2 ) , ( 4 , 3 π 2 )

To determine

To calculate: The polar equation for the ellipse with its focus at the pole and the vertices (2,π2) and (4,3π2).

Explanation

Given:

The vertices of the ellipse are (2,π2) and (4,3π2).

Formula used:

In polar coordinates, the equation of the ellipse with its focus at the pole is:

r=ed1+esinθ

Here e is the eccentricity and d is the distance between the focus and its corresponding directrix.

Calculation:

Consider the first vertex,

(r,θ)=(2,π2)

Now, substitute these values in the standard equation of the ellipse.

Then,

r=ed1+esinθ2=ed1+esin(π2)2=ed1+e

So,

ed=2(1+e) …… (1)

Now, consider the second vertex,

(r,θ)=(4,3π2)

Now, substitute these values in the standard equation of the ellipse

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