SOLUTION: Please help me find the solution & properties of ellipse and graph 4x^2=8y^2+8x-4y=8 Center: Vertices: Endpoints of minor axis: Foci:

Question 1045034: Please help me find the solution & properties of ellipse and graph
4x^2=8y^2+8x-4y=8
Center:
Vertices:
Endpoints of minor axis:
Foci:
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
It must be , or something like that.
--->--->
When you look at that you realize that you almost have two squares:
and .
So, you add to both sides of the equal sign of the original equation to get

At that point you realize that
since and only appear once,
and as and ,
the curve represented by that equation has vertical and horizontal axes of symmetry, given by
<---> and
<---> .
So, the major and minor axes must be along those lines.
That means that the center is the point with .

You also realize that the ellipse cannot stray too far from that center.
The horizontal and vertical distance from the center to a point on the ellipse,
and
have maximum possible values,
because and have maximum possible values.

No matter what value takes,
since <---> ,
,
and means .
so there are horizontal ends to the ellipse.
Similarly, no matter what value takes,
there are vertical ends to the ellipse
Since <---> ,
,
and <---> .
Since ,
the ellipse stretches farther in the horizontal) x-direction.
That means that the horizontal axis is the major axis.

The ends of the ellipse along the major axis are the vertices.
Their distance to the center is the semi-major axis, , given by
<--> .
So, the vertices have
<--> <--> and
<--> <--> ,
along with .

The minor axis, along the line , has ends at a distance from the center.
That distance is called the semi-minor axis and is given by
<---> <---> .
So, the ends of the (vertical) minor axis of symmetry of the ellipse have
and ,
along with .

The foci of the ellipse are along the major axis, ,
at a distance from the center of the ellipse,
and we know that .
Substituting the values found before,
and , we get
---> ---> ---> ---> ---> .
So, the coordinates for the foci are , along with
for one focus, and
for the other focus.
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