Write the standard equation of the ellipse with the given characteristics
Center:(0,0)
foci: (2,0) (-2,0)
vertices:(5,0) (-5,0)
looks like the vertices and the foci are on the x-axis, so this is easy
a = 5, b = ? and c = 2
In this type of ellipse,
b^2 + c^2 = a^2
b^2 + 4 = 25
b^2 = 21
x^2 /25 + y^2/21 = 1
To write the standard equation of an ellipse, we need to consider its center, foci, and vertices. Here's how you can find it using the given characteristics:
1. Center: The center of the ellipse is given as (0,0). This means that the coordinate (h, k) in the standard equation will be (0, 0). So, we have (h, k) = (0, 0).
2. Foci: The foci of the ellipse are given as (2,0) and (-2,0). The distance between the center and each focus, which we'll call c, can be found using the formula: c = distance between foci / 2. In this case, c = (2-(-2))/2 = 4/2 = 2.
3. Vertices: The vertices of the ellipse are given as (5,0) and (-5,0). The distance between the center and each vertex, which we'll call a, can be found using the formula: a = distance between vertices / 2. In this case, a = (5-(-5))/2 = 10/2 = 5.
Now that we have the values of a and c, we can write the standard equation of the ellipse in the form:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
Since the center is (0,0) (h, k = 0) and b^2 = a^2 - c^2, our equation becomes:
x^2/5^2 + y^2/b^2 = 1
To find b^2, we substitute the values of a^2 = 5^2 and c^2 = 2^2 into the equation:
b^2 = a^2 - c^2
= 5^2 - 2^2
= 25 - 4
= 21
Therefore, the standard equation of the ellipse with the given characteristics is:
x^2/25 + y^2/21 = 1