For an ellipse, one focus is (0,0), one vertex is (2,0), and the eccentricity is 0.4. How do I find the equation for the ellipse? It seems like I have so little information!
x^2/a^2 + y^2/b^2 = 1 is the standard equation , and eccentricity=sqrt(1-b^2/a^2)
Here the equation is..
second point.
4/a^2=1 a=2
first point:
.4=sqrt(1-b^2/4)
.16=1-b^2/4
.64=4-b^2
b^2=3.36
x^2/4 + y^2/3.36=1
check all that work
I get it now! However, I think you made a mistake with .16=1-b^2/4. It should be turned into .64=1-b^2, and then b^2=0.36. Then x^2/4 + y^2/0.36=1.
To find the equation for the ellipse given the focus, vertex, and eccentricity, you can follow these steps:
1. Recall that the definition of an ellipse is the set of all points such that the sum of the distances from any point on the ellipse to two fixed points, called the foci, is constant.
2. In this case, one focus is given as (0,0). Let's denote the position of the other focus as (c,0).
3. The distance between the two foci is 2c, and the eccentricity (e) of an ellipse is the ratio of the distance between the foci to the length of the major axis.
4. Since the eccentricity is given as 0.4, we can express it as e = c/a, where a is the length of the semimajor axis.
5. From the given information, we can deduce that the distance from the center (0,0) to the vertex (2,0) is equal to the length of the semimajor axis (a).
6. Using the distance formula, we have sqrt((2 - 0)^2 + (0 - 0)^2) = a, which simplifies to sqrt(4) = a, so a = 2.
7. Now, we can solve for c using the relationship between e, c, and a. With e = 0.4 and a = 2, we have 0.4 = c/2, which gives us c = 0.8.
8. The equation for the ellipse centered at the origin (0,0) with one focus at (0,0), a vertex at (2,0), and an eccentricity of 0.4 is (x^2)/2^2 + (y^2)/(2^2 + (0.8)^2) = 1.
So, the equation for the ellipse is (x^2)/4 + (y^2)/(4 + 0.64) = 1.
This is how you can find the equation of the ellipse given the provided information.