The radius of an ellipse from the origin.

In summary, the conversation discusses the use of polar coordinates to solve for the value of e in the equation (x^2)/(a^2) + (y^2)/(b^2) = 1, where x=rcos(phi) and y=rsin(phi). The hint given is to use the trigonometric identity cos^2(phi) + sin^(phi) = 1, and the category for this topic would be precalculus mathematics.
  • #1
Erez
2
0
Hello,
given (x^2)/(a^2) + (y^2)/(b^2) = 1.
and using polar coordinates x=rcos(phi) , y=rsin(phi),
equating gives r^2 = 1/[(cos^2(phi)/a^2) + (sin^2(phi)/b^2)].
or if we leave b in the nominator :
r= b/[(sin^2(phi)+(b^2/a^2)cos^2(phi)]^1/2.

-could someone give a hint as to how the demoninator of the last expression can be turned into [1 - (e^2)cos^2(phi)]^1/2 where e is the eccentricity of the ellipse?
and what is the value of e?


thank you.
 
Last edited:
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  • #2
and what is the value of e

[tex]e^2=1-\frac{b^2}{a^2}[/tex]

And by the way this is introductory physics section.
 
  • #3
Welcome to PF!

Hi Erez! Welcome to PF! :smile:

Hint: use cos^2(phi) + sin^(phi) = 1.

Then … ? :smile:
 
  • #4
Hello,
I am surprised I recived a reply so quick,
thank you.
p.s. in what category/section would this post belong ?
 
  • #6
Erez said:
Hello,
I am surprised I recived a reply so quick,
thank you.
p.s. in what category/section would this post belong ?

Hi Erez :smile:

Well, it was a short question, clearly stated, without loads of irrelevant gumph to read through … and some of us give questions like that priority! :wink:

Well, this is just geometry, so it should really have gone into "Precalculus Mathematics", which is defined as "All math courses prior to calculus" :smile:
 

1. What does the radius of an ellipse from the origin represent?

The radius of an ellipse from the origin is the distance from the center of the ellipse to any point on its perimeter. It is essentially the length of the semi-major axis of the ellipse.

2. How is the radius of an ellipse from the origin calculated?

The radius of an ellipse from the origin can be calculated using the formula r = a * b / √(a^2 * cos^2θ + b^2 * sin^2θ), where a and b are the lengths of the semi-major and semi-minor axes respectively, and θ is the angle between the semi-major axis and the line connecting the center of the ellipse to the point on its perimeter.

3. Is the radius of an ellipse from the origin constant?

No, the radius of an ellipse from the origin is not constant. It varies depending on the point on the perimeter of the ellipse that is being measured from the origin.

4. What is the relationship between the radius of an ellipse from the origin and its eccentricity?

The radius of an ellipse from the origin is inversely proportional to the eccentricity of the ellipse. This means that as the eccentricity increases, the radius decreases, and vice versa.

5. How does the radius of an ellipse from the origin affect its shape?

The radius of an ellipse from the origin plays a crucial role in determining the shape of the ellipse. In fact, the equation of an ellipse can be written in terms of its radius, where a and b are the lengths of the semi-major and semi-minor axes respectively and r is the radius from the origin. This shows that the radius is a fundamental factor in defining the shape and size of an ellipse.

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