How do we deduce that an ellipse, when defined as a “stretched circle”, has foci? [duplicate]

I've learned two separate ways to define an ellipse:

1. It's a stretched circle. We get the formula for a unit circle, $$X^2 + Y^2 = 1$$, and stretch it by dividing the terms like so: $$(\frac{X}{a})^2 + (\frac{Y}{b})^2 = 1$$. In order to satisfy the same equation, for every $$Y$$ we previously had, $$X$$ must get stretched by a factor of $$a$$, and for every $$X$$ we previously had, $$Y$$ must be stretched (multiplied) by a factor of $$b$$.

2. An ellipse is the set of all points whose sum of the distances from two points, the foci, is a constant. We can represent this with the equation $$\sqrt{(x+f)^2 + y^2} + \sqrt{(x-f)^2 + y^2} = c^2$$, where $$c = 2a$$ from the previous equation, $$f$$ is the distance from a focus to the origin, and $$x$$ and $$y$$ are the variables.

But as far as I've learned, it all seems like everything having to do with the foci in the first definition comes from first accepting that the foci DO indeed exist from the second definition. I can't seem to find a connection between them (the first definition doesn't even have an $$f$$ in it!!!).

How can we deduce that such "foci" exist from the first explanation? Every explanation of ellipses I've seen seems to use the stretched circle equation, and then go on to deduce the rest of the characteristics of the ellipse but only ASSUMING that the foci exist in the first place.

Thank you!

marked as duplicate by Henning Makholm, Blue geometry StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jan 3 at 17:32

• There's algebra that predicts where the foci should be starting from $a$ and $b$ in your first definition. Then you can check that those two points when used in the second definition give you the same ellipse. Then anything you derive about the ellipse from either definition will be valid when you use the other. – Ethan Bolker Jan 3 at 16:06