# Ellipses - Basic Concept Question

• jacksonpeeble
In summary, the conversation discussed conic sections and ellipses, including the basics of foci, semi-major and semi-minor axes. The equation c²= a² − b² was mentioned, which is similar to the Pythagorean Theorem. The conversation also mentioned deriving Cartesian coordinate equations for conic sections and provided examples for circles, ellipses, and parabolas. The recurring appearance of these curves in physical phenomena was also noted. Finally, the conversation explained the defining property of an ellipse and how it relates to the Pythagorean Theorem.
jacksonpeeble
Gold Member
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)?

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.

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 $\sqrt{c^2+ b^2}$ and the distance from (0, b) to (c, 0) is the same. Since that total distance is a constant, we have $\sqrt{c^2+ b^2}= a$ or, after squaring, $c^2+ b^2= a^2$ and $c^2= b^2- a^2$.

Thanks!

## What is an ellipse?

An ellipse is a geometric shape that resembles a flattened circle. It is defined as a closed curve with two focal points, in which the sum of the distances from any point on the curve to the two focal points is constant.

## How is an ellipse different from a circle?

An ellipse is different from a circle in that it has two focal points instead of one, and it is not symmetrical. The distance from the center of an ellipse to any point on the curve varies, whereas the distance from the center of a circle to any point on the curve is constant.

## What are the parameters of an ellipse?

The parameters of an ellipse include its major axis, minor axis, and eccentricity. The major axis is the longest diameter of the ellipse, while the minor axis is the shortest diameter. Eccentricity is a measure of how flattened or elongated the ellipse is, and it is equal to the distance between the two focal points divided by the length of the major axis.

## How can an ellipse be represented mathematically?

An ellipse can be represented mathematically using its standard equation, (x^2/a^2) + (y^2/b^2) = 1, where a is the length of the semi-major axis and b is the length of the semi-minor axis.

## What are some real-life examples of ellipses?

Ellipses can be found in many natural and man-made structures. Some examples include the orbits of planets around the sun, the shape of an egg, the outline of a hurricane, and the design of architectural structures such as bridges and arches.

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