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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

Finding a Polar Equation In Exercises 123-126, find a polar equation Tor the conk with its focus at the pole and the given vertex or vertices.

Conic Vertex or Vertices
Ellipse ( 5 , 0 ) , ( 1 , π )

To determine

To calculate: A polar equation for the conic whose focus is at the pole and the vertices are (5,0), (1,π).

Explanation

Given:

The provided conic is ellipse and the vertices are (5,0), (1,π).

Calculation:

The focus of the parabola is at the pole and the vertices of the ellipse are (5,0) and (1,π).

Firstly, convert the coordinates of the vertex into rectangular coordinates.

Consider the first pair of coordinates: (5,0)

Here, r=5 and θ=0.

So,

x=rcosθ=5cos0=5

In the same way,

y=rsinθ=5sin0=0

So, one vertex is at (5,0).

Consider the second pair of coordinates: (1,π)

Here, r=1 and θ=π.

Thus,

x=rcosθ=1cosπ=1

In the same way,

y=rsinθ=1sinπ=0

So, another vertex is at (1,0).

Therefore, the major axis of the ellipse is horizontal.

The standard polar equation of the ellipse with horizontal major axis is

r=ed1+ecosθ

Put (r,θ)=(5,0).

Then,

5=ed1+ecos(0)

Thus,

5=ed1+e. .. .. (1)

Put (r,θ)=(1,π)

Then,

1=ed1+ecos(π)

Thus,

1=ed1e

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