Friday, October 19, 2012

Here's a preview on the ellipse:

Another type of conic section is an ellipse. An ellipse can sometimes resemble a circle or will be more long and narrow, almost oval shaped. The ellipse is a set of all points in a single plane where the total distance from two fixed points is constant. This is illustrated in the picture on the next page. Notice the points a and b, known as the focus, and distances d1 and d2. If you place thumbtacks at a and b, tie a string to each and place a pen point at the loop and pull until tight, we can draw the ellipse by keeping the string tight.

The equation of the ellipse in standard form that is symmetric with both axes with a center of (0, 0) is given by  x^2 + y^2  =1.
     a^2    b^2

where a > 0 and b > 0.The intercepts of the graph are (a , 0), (-a, 0), (0, b) and (0, -b). In an ellipse that is elongated horizontally, a > b. This is easy to remember because the larger number will be under the x2 and the elongation is along the x- axis. If a < b, the ellipse is elongated vertically. This is also easy to see since the larger number will be under the y2 and the elongation is along the y- axis. The foci (plural of focus) are at (c, 0) and (-c, 0), where c2 = a2 – b2 . The vertices are the endpoints of each axis. The line segment joining the vertices of the elongated side is called the major axis and the line segment joining the vertices of the shorter side is called the minor axis. The following example will illustrate all of the points above about
the ellipse centered at (0, 0)

Graph x^2  + y^2 = 1
           25        9

We will first plot the intercepts by solving for a and b. From the standard equation of the ellipse, we know that a^2 = 25and b^2 = 9. Therefore a = 5 and b = 3. The intercepts are then (5, 0), (-5, 0), (0, 3) and (0, -3). Since a > b we know that major axis is the x- axis and the minor axis is the y- axis. We calculate the value of c from c2 = b2 – a2 to plot the foci.

Therefore c^2 = 25 – 9 = 16, so c = 4. The foci are at (4, 0) and (-4, 0). We can draw the ellipse through the 4 points or we can add more points by substituting values in for x or y. For practical purposes, there is no need to add extra points. Notice the graph of the ellipse below

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