Write the equation of the ellipse using the given information:
The ellipse has foci (4, 1) and (8, 1) and vertices (1, 1) and (11, 1);
from the foci, it is clear that the center is at (6,1) and
c = 2
Since the major axis has length 10, a=5
b^2 = 25-4 = 21
so, the equation is
(x-6)^2/25 + (y-1)^2/21 = 1
see the data at
http://www.wolframalpha.com/input/?i=ellipse+(x-6)%5E2%2F25+%2B+(y-1)%5E2%2F21+%3D+1
Well, well, well, let's get down to business and find the equation of this fancy ellipse.
First things first, we know that the center of the ellipse lies right in the middle of the two vertices. So let's calculate the center. The x-coordinate of the center is the average of the x-coordinates of the vertices, and the y-coordinate stays the same. Therefore, the center is (6, 1). Talk about finding the perfect spot!
Next up, we need to figure out the distance between the center and one of the foci. Since the foci are (4, 1) and (8, 1), both of them are at a distance of 2 units from the center along the x-axis. So the distance between the center and a focus is 2.
Now, here's the fun part. We can use a fancy formula to find the length of the major axis of the ellipse, which is denoted by 2a. The formula is:
2a = distance between the vertices
In this case, the distance between the vertices is 11 - 1 = 10. Yep, it's as simple as that! Therefore, 2a = 10, which means a = 5.
Last but certainly not least, we can determine the length of the minor axis of the ellipse, which is denoted by 2b. The length of the minor axis is the distance between the foci, so we got 2b = 8 - 4 = 4. Divide this by 2, and we get b = 2.
Now it's time to bring all this information together and write the equation of this ellipse. Here it is:
(x - 6)^2 / 5^2 + (y - 1)^2 / 2^2 = 1
Ta-da! There you have it, the equation of the ellipse that will have you spinning (not literally, of course).
To find the equation of the ellipse, we can start by finding the center of the ellipse. The center of an ellipse is halfway between the foci. Therefore, the center of this ellipse is ((4 + 8) / 2, (1 + 1) / 2) = (6, 1).
Next, we can find the length of the major axis, which is the distance between the two vertices. In this case, the length of the major axis is 11 - 1 = 10.
The distance between the center and each vertex is half the length of the major axis. Therefore, the distance is 10 / 2 = 5.
Thus, the equation of the ellipse with foci (4, 1) and (8, 1) and vertices (1, 1) and (11, 1) is:
(x - 6)^2 / 5^2 + (y - 1)^2 / b^2 = 1,
where b is the length of the minor axis. To find b, we can use the relationship between the length of the major and minor axes of an ellipse:
a^2 - b^2 = c^2,
where a is the length of the major axis and c is the distance from the center to each focus. In this case, a = 5, and c = 6 - 4 = 2. Substituting these values into the equation, we have:
5^2 - b^2 = 2^2,
25 - b^2 = 4,
b^2 = 21.
Therefore, the equation of the ellipse is:
(x - 6)^2 / 5^2 + (y - 1)^2 / 21 = 1.
To find the equation of an ellipse using the given information, we will use the standard form of the equation for an ellipse:
((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 one of the vertices, and 'b' is the distance from the center to one of the co-vertices.
Step 1: Find the center of the ellipse.
The center of the ellipse is the midpoint between the foci. In this case, the foci are (4, 1) and (8, 1). So, the x-coordinate of the center is (4 + 8) / 2 = 6, and the y-coordinate is 1. Therefore, the center of the ellipse is (6, 1).
Step 2: Find the value of 'a'.
The distance between the center and one of the vertices is 'a'. In this case, the vertices are (1, 1) and (11, 1). The difference between their x-coordinates is 11 - 1 = 10. Therefore, 'a' is half of that distance, so 'a' = 10 / 2 = 5.
Step 3: Find the value of 'b'.
The distance between the center and one of the co-vertices is 'b'. In this case, the co-vertices are also (1, 1) and (11, 1). The x-coordinate remains the same, so the difference between their y-coordinates is 0. Therefore, 'b' = 0 / 2 = 0.
Step 4: Write the equation of the ellipse.
Plug the values we found into the standard form equation:
((x - 6)^2 / 5^2) + ((y - 1)^2 / 0^2) = 1
However, since 'b' is 0, the equation simplifies to:
((x - 6)^2 / 5^2) + 0 = 1
Final equation of the ellipse:
(x - 6)^2 / 5^2 = 1
Therefore, the equation of the ellipse is (x - 6)^2 / 25 = 1.