# What is the misconception about the polar equation of a hyperbola at theta = pi?

In summary, the conversation discusses an equation of a hyperbola in polar form and the confusion over the value of R when theta is equal to pi. There is a disagreement over the correct value of R and a suggestion to use absolute value to ensure a positive value. The conversation also mentions a possible mistake in the given expression and clarifies that while the points in the xy plane may appear to be (-1,0), their coordinates in the polar coordinate system are (1,pi) due to the requirement of a positive radius.
I think that I'm over looking something with this problem. Below is the equation of an hyperbola in polar form.

$$R=\frac{1}{1 + 2cos{\theta}}$$

when $$\theta =\pi$$ shouldn't $$R = -1$$? And not $$R= 1$$
Am I over looking some property of the $$\cos$$ function?
Even when i evalute this expression at $$\theta=\pi$$in my ti-89 i get that R is = to 1. What am i not seeing?

You have made a mistake,im afraid.
Even if you find that R=-1,it is just a mathmatic form.
R is always positive,you can just say R=|1/2cos(theta)|
If still don't understand,contact me at wangkehandsome@hotmail.com,I will be glad to anwser it for you and even be more glad if you point out my fallacy.

Well, then he hasn't made a mistake: the given expression is just wrong!

I see the error. Both of you are correct. the expression is not explicit enough. when i plot these points in the xy plane they are (-1,0) BUT the same coordinates in the
$$(r,\theta)$$ coordinate system are $$(1,\pi)$$ because the radius is always positive

## 1. What is the difference between polar and Cartesian coordinates?

In polar coordinates, a point is represented by a distance from the origin (called the radius) and an angle from a fixed reference line (usually the positive x-axis). In Cartesian coordinates, a point is represented by its distance from the x-axis (called the x-coordinate) and its distance from the y-axis (called the y-coordinate).

## 2. How do you convert a polar equation to Cartesian form?

To convert a polar equation to Cartesian form, use the following substitutions: x = r cos θ and y = r sin θ. Then use trigonometric identities to eliminate the variable r and simplify the equation.

## 3. What are some common conic shapes in polar equations?

The most common conic shapes in polar equations are circles (r = a), ellipses (r = a(1 ± e cos θ)), parabolas (r = a(1 ± cos θ)), and hyperbolas (r = a(1 ± e sec θ)).

## 4. How do you graph a polar equation?

To graph a polar equation, plot points by substituting various values of θ into the equation and finding the corresponding values of r. Then connect the plotted points to create the desired shape.

## 5. What is the focus-directrix property of conic sections in polar form?

In polar form, the focus-directrix property states that the distance from any point on a conic section to the focus is equal to the distance from that same point to the corresponding directrix line. This property is useful for finding the eccentricity and key points of a conic section.

• Introductory Physics Homework Help
Replies
8
Views
319
• Introductory Physics Homework Help
Replies
2
Views
674
• Introductory Physics Homework Help
Replies
1
Views
394
• Introductory Physics Homework Help
Replies
25
Views
2K
• Introductory Physics Homework Help
Replies
10
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
925
• Introductory Physics Homework Help
Replies
1
Views
160
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
99
Views
8K