Are Focal Radii Equal to 2a in an Ellipse?

[xLaser]

In an ellipse, with center (0,0) can you assume the focal radii to be equal to 2a? where 2a is the length of the major axis?

how about with center (h,k)?

I'm pretty sure I read before that u cannot assume it to be 2a in an ellipse, only in hyperbola's. BUt my teacher tells me otherwise.

Please advice,

thx.

 

[dextercioby]

What's the equation for an ellipse which has the center in the point (h,k) and semiaxis (a,b)??

Daniel.

 

[BobG]

In an ellipse, with center (0,0) can you assume the focal radii to be equal to 2a? where 2a is the length of the major axis?

how about with center (h,k)?

I'm pretty sure I read before that u cannot assume it to be 2a in an ellipse, only in hyperbola's. BUt my teacher tells me otherwise.

Please advice,

thx.

I assume you're talking about the sum of the distance from each focus to the ellipse. If so, your teacher is correct.

In fact, that's the easiest way to draw an ellipse. Thumbtack a piece of paper to a piece of cardboard with the thumbtacks somewhere near the center of the paper. Place a loop of string around the tacks. Pull the loop taut with your pencil and start drawing, always keeping your string taut. Since the distance between the tacks isn't changing and the length of the loop isn't changing, you can be sure the length from one focus to the pencil plus the length from the pencil to the other focus remains constant.

How do you know that's actually equal to the major axis (2a)? Put your pencil at one of the vertexes. You have your distance from the far focus plus the distance from the near focus. If the distance from the near focus to the ellipse is the same as when you repeat this exercise on the opposite side of the ellipse, then the total distance must be equal to the length of the major axis. The length of the string is constant and the distance between the tacks is constant, so it must give the same length regardless of which vertex you chose to do this on.

Hopefully, that comes across in words. If you actually draw the ellipse using the loop and tacks and play with it a little, it becomes pretty obvious why your teacher's correct even without any words.