Find Length of Line Segment Connecting Center & Point of Ellipse

[32l3n]

Hello to everyone!
I am really desparately trying to find the length of the "radius" of an ellipse.I will explain exactly what I mean by that - its the length of the line segment that connects the center of a particular ellipse with a given point of the same ellipse.All the information I have to solve this problem is the ellipse equation and an angle measured from the major axis to that line segment,the center of the ellipse and cartesian coordinate system .
All kinds of tips and hints will be greatly appreciated.

 

[A. Bahat]

Well, suppose (for simplicity) that the ellipse is centered at the origin and has major axis $a$ and minor axis $b$, so that the equation of the ellipse in Cartesian coordinates is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$. I take it you want to find the equation of the ellipse in polar coordinates, i.e. find $r$ as a function of $\theta$. In that case, just do what you always do to convert to polar coordinates: plug in $x=r\cos\theta$ and $y=r\sin\theta$.

 

[32l3n]

Well, suppose (for simplicity) that the ellipse is centered at the origin and has major axis $a$ and minor axis $b$, so that the equation of the ellipse in Cartesian coordinates is $(\frac{x}{a})^2+(\frac{y}{b})^2=1$. I take it you want to find the equation of the ellipse in polar coordinates, i.e. find $r$ as a function of $\theta$. In that case, just do what you always do to convert to polar coordinates: plug in $x=r\cos\theta$ and $y=r\sin\theta$.

Thank you for the detailed answer, A.Bahat.It looks like this will solve the mystery.I don't understand the polar coordinate system and that's why I skiped this as an option.As I understand this polar equation represents a vector with a beginning the center of the Ellipse and an arrow which points at a specific fragment of the ellipse depending on the angle.

 

[A. Bahat]

That's a pretty good way to put it. The Cartesian coordinates give you the horizontal and vertical distances from the $x$ and $y$ axes (these are just the $x$ and $y$ coordinates, after all). On the other hand, polar coordinates specify the distance from the origin (called $r$, as in radius) and the angle that the line through the point makes with the $x$ axis (usually denoted by $\theta$). It is pretty easy to go back and forth between these coordinate systems using some trigonometry. This picture shows how one finds the formulas I gave in the last post for $x$ and $y$ in terms of $r$ and $\theta$.

That's really all there is to polar coordinates (along with converting polar coordinates to Cartesian coordinates, but that doesn't have anything to do with your question).

 

[blue_raver22]

Hello there,

Thank you for reaching out for help with your problem. I am happy to assist you in finding the length of the line segment connecting the center and a point on an ellipse.

Firstly, it is important to understand the equation of an ellipse. An ellipse can be described by the equation:

(x/a)^2 + (y/b)^2 = 1

where 'a' and 'b' are the semi-major and semi-minor axes, respectively. The center of the ellipse is at the origin (0,0) and the major axis is along the x-axis.

To find the length of the line segment connecting the center and a point on the ellipse, we need to know the coordinates of the point and the angle it makes with the major axis.

Let the point on the ellipse be (x1, y1) and the angle it makes with the major axis be 'θ'. Using trigonometry, we can find the length of the line segment as:

r = √(x1^2 + (y1/b)^2) + (a*cosθ)^2 + (b*sinθ)^2)

where 'r' is the length of the line segment.

I hope this helps you in solving your problem. If you need further assistance, please feel free to reach out to me.

Best regards,