Finding the Equation of an Ellipse

[CeceBear]

Homework Statement

Find the standard form of the equation of the ellipse with vertices (4,0) and (-2,0) and foci (3,0) and (-1,0).

Homework Equations

((x-h)^2/a^2) + ((y-k)^2/b^2) = 1

c^2 = a^2 + b^2

The Attempt at a Solution

I haven't got much of anything:
(x)^2 / 8
I'm pretty sure this is wrong. I have no problem finding the foci and vertices if given the equation, but I can't seem to understand how to do it the other way around. Is the center (0,0)?

 

[Mark44]

Homework Statement

Find the standard form of the equation of the ellipse with vertices (4,0) and (-2,0) and foci (3,0) and (-1,0).

Homework Equations

((x-h)^2/a^2) + ((y-k)^2/b^2) = 1

c^2 = a^2 + b^2

The Attempt at a Solution

I haven't got much of anything:
(x)^2 / 8

Right, this isn't much of anything. For one thing, it isn't an equation.

I'm pretty sure this is wrong. I have no problem finding the foci and vertices if given the equation, but I can't seem to understand how to do it the other way around. Is the center (0,0)?

No.
The first thing you should do is to draw a rough sketch of the ellipse using the given information about the two vertices and the foci. These alone should convince you that the center is not at the origin.

In your equation c² = a² + b², what do the letters a, b, and c represent? From your sketch can you figure out what two of these constants need to be?

 

[CeceBear]

Well, I figured out the center should be (1,0), right?
I can't fully sketch the ellipse without knowing the equation.

If the vertices are: (-a, 0) (a, 0)
and the foci are: (-c,0) (c,0)
where c^2 = a^2 + b^2

Then the equation would be: ((x-1)^2 / a^2) + (y^2 / b^2)

Which vertice and focal point do I use to find the a, b, and c values?

 

[Mark44]

Well, I figured out the center should be (1,0), right?

Right.

I can't fully sketch the ellipse without knowing the equation.

It doesn't need to be very accurate, but put the vertices and foci where they belong.

If the vertices are: (-a, 0) (a, 0)
and the foci are: (-c,0) (c,0)
where c^2 = a^2 + b^2

Isn't c the distance from the center to either focus?

Then the equation would be: ((x-1)^2 / a^2) + (y^2 / b^2)

That's not an equation since it doesn't have an equals sign.

Which vertice and focal point do I use to find the a, b, and c values?

a is the distance from the center to either vertex (vertice is not a word), assuming that the major axis is horizontal rather than vertical.

 

[CeceBear]

Would the equation be:

((x-1)^2 / 16) + ((y^2) / 7) = 1

 

[SteamKing]

You're just guessing. If you would research the term "ellipse" on the internet, you would obtain the answers to your questions easily.

 

[CeceBear]

You're just guessing. If you would research the term "ellipse" on the internet, you would obtain the answers to your questions easily.

That wasn't just a guess. I'm trying to figure this out based on the notes and the process my teacher showed me. But obviously that's not working...

 

[Mark44]

If the vertices are: (-a, 0) (a, 0)
and the foci are: (-c,0) (c,0)
where c^2 = a^2 + b^2

This isn't correct. In the third equation, c has to be larger than a, but the foci are inside the ellipse. Check your book for the right formulas.