In a previous section we looked at graphing circles and since circles are really special
cases of ellipses we’ve already got most of the tools under our belts to
graph ellipses. All that we really need here to get us started is then standard form of the
ellipse and a little information on how to interpret it.
Here is the standard form of an ellipse.
Note
that the right side MUST be a 1 in order to be in standard form. The point (h, k
) is called the center
of the ellipse.
To
graph the ellipse all that we need are the right most, left most, top most and
bottom most points.
Once we have those we can sketch in the ellipse. Here are formulas for finding
these points.
right most
point : (h + a, k )
top most point
: (h, k + b )
bottom most point
: (h, k − b )
Note
that a is the square root of the number under the x term and is
the amount that we move right
and left from the center. Also, b is the square root of the number under
the y term and is the amount that we move up or down from the center.
Finally,
let’s address a comment made at the start of this section. We said that circles
are really nothing
more than a special case of an ellipse. To see this let’s assume that a = b . In
this case we
have,
Note
that we acknowledged that a = b and used a in both cases. Now if we clear denominators we
get,
(x − h)2 + ( y − k )2 = a2
This
is the standard form of a circle with center (h, k ) and radius a.
So, circles really are special cases
of ellipses.
Sumber
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