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Chapter 10: Problem 51

Write a polar equation of the conic that is named and described. Ellipse: a focus at the pole; vertex: \((4,0) ; e=\frac{1}{2}\)

Short Answer

Expert verified
The polar equation is \( r = \frac{4}{1 - \frac {cos(\theta)}{2} } \)

Step by step solution

01

Identify Parameters

The eccentricity, e, is given directly as \( e = \frac{1}{2} \). The vertex is the point (4,0), so the distance from the focus to the vertex (the semi-major axis), denoted by a, is 4.
02

Compute the distance from the focus to the directrix, p

In an ellipse with a focus at the origin, the relationship between a, e, and p is given by \( a = \frac {ep}{1-e} \). We know a and e, so we can solve for p: \( a = \frac {ep}{1-e} \) becomes \( 4 = \frac {(\frac{1}{2})p}{1 - \frac{1}{2}} \) which simplifies to \( 4 = \frac {p}{2} \) or \( p = 8 \)
03

Write the Polar Equation

Substitute the values of e and p into the general equation \( r = \frac{ep}{1 - ecos(\theta)} \) this gives \( r = \frac{( \frac{1}{2} * 8 )}{ 1 - ( \frac{1}{2} * cos(\theta) ) } \), simplifying to \( r = \frac{4}{1 - \frac {cos(\theta)}{2} } \)

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Most popular questions from this chapter

Where possible, find each product. a. \(\left[\begin{array}{rr}{1} & {0} \\ {0} & {-1}\end{array}\right]\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right]\) b. \(\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right]\left[\begin{array}{rrr}{-1} & {0} & {1} \\ {0} & {-1} & {1}\end{array}\right]\)

If all conics are defined in terms of a fixed point and a fixed line, how can you tell one kind of conic from another?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so 1 was able to graph the ellipse.

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$ r=\frac{12}{2+4 \cos \theta} $$

Use the Law of sines to solve triangle \(A B C\) if \(A=35^{\circ}, a=11,\) and \(b=15 .\) Assume \(B\) is acute. Round lengths of sides to the nearest tenth and angle measures to the nearest.
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