Ellipses - Basic Concept Question

jacksonpeeble

In class today, my instructor went over conic sections and ellipses (and hyperbolas, although that's irrelevant). We pretty much learned the basics - foci, semi-major and semi-minor axes, etc.

However, the equation c²= a² − b² where c is the distance from the focus to vertex and b is the distance from the vertex a co-vetex on the minor axis sort of confused me. It sounds a lot like the Pythagorean Theorem. Could someone please explain why this formula is true (graphically)?

slider142

You'll have a greater handle on conic sections by deriving the Cartesian coordinate equations yourself from their geometric definitions. If you're feeling adventurous, you can try the intersection of a plane with two cones. Otherwise, try these:
1) A circle is the set of all points equidistant from a single point. Suppose this point is given the Cartesian coordinate (h, k). If (x, y) is a point on the circle, what equation must x, y, h, and k satisfy? Note that since the circle is defined by distance, the definition of Euclidean distance, the Pythagorean equation, will be necessary.
2) An ellipse is the set of all points (x, y) such that the sum of the distances from two particular points (called focii) in the plane is a constant L. This is like attaching two thumbtacks to a sheet of paper and attaching a string of length L between them, then using a pencil to draw the shape that always keeps the string taut.
3) A parabola is the set of all points in the plane equidistant from a particular point, called the focus, and a line, called the directrix (the distance between a point and a line is taken as the minimum distance). You can also try to get it as the shape from which all lines perpendicular to the directrix are reflected by the curve out through the focus.
You will find all of these curves and the hyperbola recurring many times in physical phenomena, so their properties should be second nature.

HallsofIvy

An ellipse has the "defining" property that there are two points, the foci, such that the total distance from one focus to any point on the ellipse to the other focus is a constant.

Suppose the foci are at (-c, 0) and (c, 0) and the ellipse crosses the x-axis at (a, 0). Going from (-c, 0) to (a, 0) is a distance of (a+ c): from the focus to the origin is c and from the origin to the ellipse is a. Now back to the focus is a distance of a- c: we only go back to (c, 0), not to the orgin or (-c, 0). The (constant) total distance is (a+c)+ (a- c)= 2a.

Now suppose the ellipse crosses the y-axis at (0, b). The total distance from (-c,0) to (0,b ) to (c,0) is the sum of two hypotenuses or right triangles: the first with vertices (-c,0), (0,0) and (0,b), the other with vertices (c, 0), (0, 0), and (0,b).

That's where the Pythagorean theorem comes in! The distance from (-c,0) to (0, b) is [itex]\sqrt{c^2+ b^2}[/itex] and the distance from (0, b) to (c, 0) is the same. Since that total distance is a constant, we have [itex]\sqrt{c^2+ b^2}= a[/itex] or, after squaring, [itex]c^2+ b^2= a^2[/itex] and [itex]c^2= b^2- a^2[/itex].

jacksonpeeble

Thanks!