Right, this isn't much of anything. For one thing, it isn't an equation.CeceBear said: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
No.CeceBear said: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)?
Right.CeceBear said:Well, I figured out the center should be (1,0), right?
It doesn't need to be very accurate, but put the vertices and foci where they belong.CeceBear said:I can't fully sketch the ellipse without knowing the equation.
Isn't c the distance from the center to either focus?CeceBear said:If the vertices are: (-a, 0) (a, 0)
and the foci are: (-c,0) (c,0)
where c^2 = a^2 + b^2
That's not an equation since it doesn't have an equals sign.CeceBear said:Then the equation would be: ((x-1)^2 / a^2) + (y^2 / b^2)
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 said:Which vertice and focal point do I use to find the a, b, and c values?
SteamKing said:You're just guessing. If you would research the term "ellipse" on the internet, you would obtain the answers to your questions easily.
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.CeceBear said:If the vertices are: (-a, 0) (a, 0)
and the foci are: (-c,0) (c,0)
where c^2 = a^2 + b^2
An ellipse is a geometric shape that resembles a flattened circle. It is defined as the set of all points in a plane where the sum of the distances from two fixed points (called foci) is constant.
The standard form of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively. To find the equation, you need to know the coordinates of the center, the lengths of the axes, and whether the ellipse is horizontal or vertical.
A circle is a special case of an ellipse where both foci are at the same point, making the two axes equal in length. This means that the standard form of a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius. An ellipse, on the other hand, has two distinct foci and its axes are of different lengths.
The eccentricity of an ellipse, denoted by e, is a measure of how elongated the ellipse is. It is defined as the ratio of the distance between the foci to the length of the semi-major axis. The closer the eccentricity is to 0, the more circular the ellipse is, and the closer it is to 1, the more eccentric the ellipse becomes.
No, an ellipse cannot have a negative eccentricity. The eccentricity is always a positive number because it is a ratio of distances, and distances cannot be negative. However, an ellipse can have an imaginary eccentricity if it is a non-real ellipse, meaning its foci are complex numbers. This is not a common occurrence and is usually only seen in theoretical mathematics.