The major axis of the ellipse lies on the line y=3 and has length 6; the minor axis lies on the line x=–4 and has length 4;
Clearly the center is at (-4,3)
That means you have an ellipse with
a = 6/2 = 3
b = 4/2 = 2, so its equation is
(x+4)^2/3^2 + (y-3)^2/2^2 = 1
Well, this ellipse seems to have a very asymmetrical personality! Its major axis is like the joker, stretching across the stage with a length of 6, while its minor axis is more of a shy introvert, hiding along the sidelines with a length of 4. It's like the comedy duo of the coordinate plane!
To find the equation of the ellipse, we need to determine its center coordinates and the lengths of its major and minor axes.
1. Center coordinates:
The center of the ellipse is at the point (h, k), where h is the x-coordinate and k is the y-coordinate.
Since the minor axis lies on the line x = -4, the x-coordinate of the center is -4.
Since the major axis lies on the line y = 3, the y-coordinate of the center is 3.
So, the center coordinates are (-4, 3).
2. Major axis length:
The major axis has a length of 6, which means the distance from the center to one of the endpoints of the major axis is 6/2 = 3 units.
Since the center coordinates are (-4, 3), one endpoint of the major axis will be at (-4, 3 + 3) = (-4, 6).
So, the major axis length is 2 times the distance from the center to one of its endpoints, which is 2 * 3 = 6 units.
3. Minor axis length:
The minor axis has a length of 4, which means the distance from the center to one of the endpoints of the minor axis is 4/2 = 2 units.
Since the center coordinates are (-4, 3), one endpoint of the minor axis will be at (-4 + 2, 3) = (-2, 3).
So, the minor axis length is 2 times the distance from the center to one of its endpoints, which is 2 * 2 = 4 units.
Therefore, the equation of the ellipse can be determined using the formula:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1,
where (h, k) is the center coordinates, a is the major axis length divided by 2, and b is the minor axis length divided by 2.
Plugging in the values we found:
((x - (-4))^2 / (6/2)^2) + ((y - 3)^2 / (4/2)^2) = 1.
Simplifying, we have:
((x + 4)^2 / 3^2) + ((y - 3)^2 / 2^2) = 1.
Therefore, the equation of the given ellipse is:
((x + 4)^2 / 9) + ((y - 3)^2 / 4) = 1.
To find the equation of the ellipse based on the given information, we need to understand the general equation of an ellipse.
The general equation of an ellipse centered at (h, k) with a major axis of length 2a and a minor axis of length 2b is given by:
((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1
In this case, we are given the following information:
1. The major axis lies on the line y = 3 and has a length of 6.
2. The minor axis lies on the line x = -4 and has a length of 4.
Let's break down the information and find the values for the equation.
1. Major axis:
The major axis lies on the line y = 3, which means the ellipse is centered at (h, 3), where h is the x-coordinate of the center.
The length of the major axis is 6, which means the distance from the center to each end of the major axis is half of the length, so a = 6/2 = 3.
Therefore, the equation becomes: ((x - h)^2 / 3^2) + ((y - 3)^2 / b^2) = 1.
2. Minor axis:
The minor axis lies on the line x = -4, which means the ellipse is centered at (-4, k), where k is the y-coordinate of the center.
The length of the minor axis is 4, which means the distance from the center to each end of the minor axis is half of the length, so b = 4/2 = 2.
Substituting this into the previous equation, we get: ((x - (-4))^2 / 3^2) + ((y - 3)^2 / 2^2) = 1.
Simplifying, we get: ((x + 4)^2 / 9) + ((y - 3)^2 / 4) = 1.
Thus, the equation of the ellipse is ((x + 4)^2 / 9) + ((y - 3)^2 / 4) = 1.