Foci (0,-3)(0,3)
Vertices (0,-4)(0,4)
Find the standard form
Foci: (0,-3) (0,3).
Vertices: (0,-4) (0,4).
The X-coordinate of the Foci and Vertices are constant at zero while the
Y-coordinates varies. Therefore, we have a y-Elipwse. So a^2 will be located under Y^2:
X^2/b^2 + Y^2/a^2 = 1.
a^2 = 4^2 = 16.
F^2 = a^2 - b^2 = 3^2,
16 - b^2 = 9,
b^2 = 7.
b = sqrt7.
Eq: X^2/7 + y^2/16 = 1.
To find the standard form of the equation of an ellipse, we need to determine the center, major and minor axes' lengths, and the orientation of the ellipse.
Given the information:
Foci: (0, -3) and (0, 3)
Vertices: (0, -4) and (0, 4)
The center of the ellipse is the midpoint between the foci and vertices. In this case, the center is (0, 0).
The major axis of the ellipse is the line segment connecting the vertices. Since the vertices are on the y-axis, the major axis is vertical.
The length of the major axis (2a) is the distance between the vertices. In this case, it is 8 units (4 units above and 4 units below the center).
The distance between the center and each focus (c) is 3 units.
Now we can use the standard form of an ellipse equation:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
Where:
(h, k) is the center of the ellipse
a is the distance from the center to either vertex on the major axis
b is the distance from the center to the endpoints on the minor axis
For our case, the equation becomes:
(x - 0)^2/4^2 + (y - 0)^2/b^2 = 1
Since the major axis is vertical, the equation becomes:
x^2/16 + y^2/b^2 = 1
To determine the value of b^2, we can use the Pythagorean theorem:
a^2 = b^2 + c^2
Substituting the given values:
4^2 = b^2 + 3^2
16 = b^2 + 9
b^2 = 16 - 9
b^2 = 7
Now we can substitute b^2 back into the equation:
x^2/16 + y^2/7 = 1
The standard form of the equation of the given ellipse is:
x^2/16 + y^2/7 = 1