SOLUTION: Determine the center, vertices, and foci for the following ellipse 18x^2+4y^2-108x+16y=106 Write word or phrase that best completes each statement or answers question

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Question 746555: Determine the center, vertices, and foci for the following ellipse 18x^2+4y^2-108x+16y=106
Write word or phrase that best completes each statement or answers question
Found 2 solutions by MathLover1, KMST:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

....complete the square

.......factor out from first group and from second group






...round to whole number

....=> and

the center is at: (,)
vertices: since the term has the larger denominator, the ellipse is and
vertices (, ≅ (, ) and (,)
and foci:
(, ≅ (, ≅
(, ) and (,)




Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
An equation of the form
represents an ellipse centered at (h,k), with an axis of length parallel to the x-axis,
and an axis of length parallel to the y-axis.
If we can transform the equation given into such a form, we will be able to find everything the problem asks for.

--> --> -->
At this point, you look at the two expressions in brackets and have to realize that we can add something to each expression to "complete a square"
is part of and
is part of
So and

So we go back to the original equation, and add to both sides of the equal sign:
--> --> --> -->
Dividing both sides of the equal sign by the equation turns into


That is the equation of an ellipse with at (3,-2).
The axis parallel to the y-axis (along ) is longer, and it is called the major axis.
Half of its length (called the semi-major axis) is

The are the ends of the major axis, at a distance from the center, and are at
(,) and (,)

The other axis is called the minor axis.
It is along the line , parallel to the x-axis.
The ends of the minor axis (often called co-vertices) are at distance

That distance is called the semi-minor axis.

An ellipse has two located on the major axis, between the center and the vertices, at a distance from the center called the focal distance. That distance , and the semi-minor axis are the legs of a right triangle with the semi-major axis for a hypotenuse.
Applying the Pythagorean theorem, we find that
--> --> --> --> --> -->
So the are at
(,) and (,)
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