The ellipse has foci (0, 3) and (0, –3) and vertices (0, 5) and (0, –5);
Well, it seems like this ellipse is having quite the identity crisis. I mean, having two foci and two vertices? It's like it's trying to be both an introvert and an extrovert at the same time! But hey, who am I to judge? We all have our quirks. Just be yourself, ellipse!
To find the equation of the ellipse, we need to determine the coordinates of the center, as well as the lengths of the major and minor axes.
1. The center of the ellipse is the midpoint between the foci, which can be found by averaging their x-coordinates and y-coordinates:
Center = ((0 + 0) / 2, (3 + (-3)) / 2) = (0, 0).
2. The major axis is the line segment connecting the vertices. Since the vertices have the same x-coordinate, the length of the major axis is the difference between their y-coordinates:
Length of the major axis = 5 - (-5) = 10.
3. The minor axis is perpendicular to the major axis and passes through the center. Since the center has the same x-coordinate as the vertices, the length of the minor axis is the difference between the x-coordinates of the foci:
Length of the minor axis = 0 - 0 = 0.
Now we have the information needed to write the equation of the ellipse:
The standard equation of an ellipse centered at (h, k) with major axis length 2a and minor axis length 2b is:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1.
In this case, the center is (0, 0), the major axis length is 10, and the minor axis length is 0. The equation of the ellipse is then:
(x-0)^2/5^2 + (y-0)^2/0^2 = 1.
Simplifying, we get:
x^2/25 + y^2/0 = 1.
Note that the denominator of the y-term is 0, which indicates that the ellipse only extends along the x-axis. Therefore, it is not a true ellipse but rather a vertical line segment with length 10.
To understand the properties of an ellipse with the given foci and vertices, we need to determine the equation of the ellipse. The general equation of an ellipse centered at the origin is:
(x^2 / a^2) + (y^2 / b^2) = 1
where "a" is the semi-major axis (half of the length between the vertices) and "b" is the semi-minor axis (half of the length between the co-vertices).
We know that the foci are located at (0, 3) and (0, -3). The distance from the center to each focus is called the "c" value. In our case, c = 3.
We can find "a" using the distance between the center and one of the vertices, which in this case is 5. Therefore, a = 5.
Now, we can find "b" using the Pythagorean theorem:
c^2 = a^2 - b^2
3^2 = 5^2 - b^2
9 = 25 - b^2
b^2 = 25 - 9
b^2 = 16
b = 4
Now that we have the values of a and b, we can write the equation of the ellipse:
(x^2 / 5^2) + (y^2 / 4^2) = 1
(x^2 / 25) + (y^2 / 16) = 1
Therefore, the equation of the ellipse is (x^2 / 25) + (y^2 / 16) = 1.
the center is at (0,0) and we have a^2 = b^2+c^2
a=5, c=3, so b=4 and the equation is
x^2/25 + y^2/9 = 1