Example 5

At this stage of our conics development, we really have not dealt with the equation of a conic in non-standard form. Recall that this is, however, a valid equation for a conic and it happens to be an ellipse. Our first goal is to rewrite this equation into standard form and then to interpret this equation as we sketch the graph. The technique involved in rewriting this equation into standard form is known as “completing the square.”

We see that

is equivalent to

Now we want to fill in the apparent gaps that have been inserted in the parentheses above. This “filling in” is completing the square. We want to write in the number that will make each set of parentheses a perfect square. We do that now:

Note that, when we add 9 to the left-hand side of the equation, we must also add it to the right-hand side. Also, we are not really adding 1 to the left-hand side; we are really adding 4 since we multiply the 1 by the 4 that is outside the parentheses. Hence, we must also add 4 to the right-hand side.

Rewriting our equation now yields

or

and we have successfully transformed the equation originally given to us into the standard equation of an ellipse. This ellipse has center and has values

and

As noted in previous examples, because a is in the denominator of the term involving the variable x, we know that the major axis of this ellipse is horizontal (parallel with the x-axis).

Note that

which is approximately 0.8660. Because this number is somewhat close to 1, we might expect a fairly flat ellipse (as opposed to a roundish one). Check this as you look at the graph of this ellipse, which is given in Figure E8.