Ellipses

    In general, an ellipse is a type of conic section that can be generated if a conical surface is cut with a plane which does not intersect the cone's base, then intersection of the cone and plane is an ellipse.  In this topic we define an ellpise as locus of points where the sum from any point on the curve to two fixed points (the foci) is a positive constant. We illustrate the definition with several examples including how to find an equation of an ellipse given some geometric information. Conversely, we also show how to find the vertices, major axis, minor axis, eccentricity, and focus given the equation of the ellipse.  

Definition (Ellipse)  An ellipse is the set of points in a plane, the sum of whose distances from two fixed points (the foci) in the plane is a positive constant. The midpoint of the segment between the foci is called the center of the ellipse. The line through the foci determines two points of the ellipse called the vertices of the ellipse. The segment from the vertices is called the major axis of the ellipse and the segment perpendicular to the major axis and incident with the ellipse is called the minor axis of the ellipse.

Proposition (Ellipse Centered at the Origin)  

    (i) The graph of with is an ellipse centered at the origin with major axis of length and minor axis of length The foci are where
    
    (ii) The graph of with is an ellipse centered at the origin  with major axis of length and minor axis of length The foci are where

Proposition (Ellipse Equation)  

    (i) The graph of with is an ellipse centered at with major axis of length and minor axis of length The foci are where
    
    (ii) The graph of with is an ellipse centered at with major axis of length and minor axis of length The foci are where

Example (Find the vertices, directrix, and foci of the ellipse)  For each of the following equations of the ellipse, find the vertices and foci.

(a)

    Solution. We have, and so we have the form The length of the major axis is 10, the length of the minor axis is and since the foci are The vertices are

(b)

    Solution. By dividing by , and trying to obtain the form we have



Thus, and The length of the major axis is , the length of the minor axis is and since the foci are The vertices are

(c)

    Solution. First we complete the square in and :















Therefore, is Continuing to try for the form we have,







Therefore, the center is Now we see that and Thus, The vertices of the ellipse are and the foci are   

Definition (Eccentricity of an Ellipse)  The eccentricity of an ellipse is ratio of the distance from the center to the focus and the distance from the center to the vertex.

    The eccentricity measures the roundness of an ellipse. If for example, the eccentric of an ellipse is 1 then the ellipse is very oval shaped. If the eccentricity is 0 then the ellipse is a circle. These examples can be seen from a formula to compute the eccentric of an ellipse.
    The eccentricity of an ellipse can be found if a standard form equation of an ellipse is given such as:

      or       ,

then the eccentricity of the ellipse is

Example (Find the Equation of the Ellipse)  Find an equation of the ellipse that has center at the origin and satisfies the geometric conditions:

(a) The vertices are and the foci are

    Solution. We have and Since we find Thus an equation of the ellipse is

(b) The vertices are and passing through

    Solution. We have An equation of the ellipse is To find we use the given point to obtain We find Therefore, an equation of the ellipse is

(c) The -intercepts are and the -intercepts are .

    Solution. We have and Thus which simplies to
    
(d) The eccentricity is and the vertices are .

    Solution. Since the vertices are we see that and so means Using we have and so an equation of the ellipse is

(e) The eccentricity is the vertices are on the -axis, and the ellipse passing through

    Solution. Since we have Also since the ellipse passes through we have Thus we need to solve the system:
    


The solutions yield and Therefore, an equation of the ellipse is

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