Direction: Using the information given on each item, write the
standard form equation of each ellipse.
1. Vertices: (12, -4), (-14, -4)
Foci: (11, -4), (-13, -4)
2. Vertices: (6,22), (6, -4)
Foci: (6, 14), (6,4)
3. Foci: (0,9), (-10,9)
Co-vertices: (-5, -21), (-5, -3)
4. Foci: (5,-8), (5,-8)
Co-vertices: (0,4), (0, -20) 5. Center: (6, 2)
Vertex: (6, -4) Co-vertex: (9,2)
1.
The y-coordinates of the vertices and foci are the same, so the major axis is parallel to the x-axis. Thus, the equation of the ellipse will have the form;
First, we identify the center. The center is halfway between the vertices, (12, -4), (-14, -4). Applying the midpoint formula, we have:
Next, we find a². The length of the major axis, 2a, is bounded by the vertices. We solve for a by finding the distance between the x-coordinates of the vertices.
a² = 169
Now we find c². The foci are given by (k±c, h). So, (k−c, h) = (11, −4) and (k+c, h) = (−13, -4). We substitute k = −3 using either of these points to solve for c.
c² = 144
Next, we solve for b² using the equation c² = a² − b².
2.
The x-coordinates of the vertices and foci are the same, so the major axis is parallel to the y-axis. Thus, the equation of the ellipse will have the form;
(h, k) which is the midpoint is;
(h, k) = (6, 9)
Next, we find a². The length of the major axis, 2a, is bounded by the vertices. We solve for a by finding the distance between the y-coordinates of the vertices.
3.
The equation of an ellipse is for a horizontally oriented ellipse and for a vertically oriented ellipse.
where (h, k) is the center and the distance c from the center to the foci is given by . a is the distance from the center to the vertices and b is the distance from the center to the co-vertices.
The center of the ellipse is half way between the vertices. Thus, the center (h, k) of the ellipse is (0, 0) and the ellipse is vertically oriented.
a is the distance between the center and the vertices, so a = c is the distance between the center and the foci, so c = 4
4.
The equation of an ellipse is
where (h, k) is the center and the distance c from the center to the foci is given by . a is the distance from the center to the vertices and b is the distance from the center to the co-vertices.
The center of the ellipse is half way between the vertices. Thus, the center (h, k) of the ellipse is (0, 0) and the ellipse is vertically oriented.
i.e h = 0, k = 0
a which is the x axis point is 4, while b is 20
The equation becomes
5.
Center: (6, 2) Vertex: (6, -4) Co-vertex: (9,2)
from the image above,
vertices = (6, -4), (6, 8) => -4+12
covertices = (9, 2) | (3, 2) => 9-3-3
semimajor axis length = 6
semiminor axis length = 6/3 = 2
h = 6
k = 2
the equation becomes;
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