write an equation of the ellipse with foci at (0,+11) and verticies at (0,+12)
The center of the ellipse is halfway between the foci and is at (0,11.5). The distance from the center to the vertex is a=0.5. The distance from the center to each focus is c=0.5.
The equation of the ellipse is:
(x-0)^2/0.25 + (y-11.5)^2/1 = 1
Simplified:
x^2/0.25 + (y-11.5)^2 = 1
AAAaannndd the bot gets it wrong yet again!
Since when does an ellipse have its center midway between its focus and its vertex?
Of course, the problem was poorly written. It should have read
write an equation of the ellipse with foci at (0,±11) and vertices at (0,±12)
Then we have
a = 12
c = 11
b^2 = 23
and the equation is
x^2/23 + y^2/144 = 1
My apologies for the mistake. Thank you for correcting it. You are correct, the center of the ellipse is at (0,0) and the semi-major axis is 12. The semi-minor axis can be found using the formula b^2 = c^2 - a^2, where c is the distance from the center to the focus. So, b^2 = 11^2 - 12^2 = 23.
Therefore, the equation of the ellipse is
x^2/23 + y^2/144 = 1.
To write an equation for the ellipse with foci at (0, +11) and vertices at (0, +12), we need to determine the key properties of the ellipse: the center, major axis length, and minor axis length.
The foci of an ellipse are points inside the ellipse that help determine its shape. The distance from the center of the ellipse to each focus is denoted by "c".
The vertices of an ellipse are the endpoints of the major axis. The distance from the center of the ellipse to each vertex is denoted by "a", which represents the major radius.
In this case, the foci are at (0, +11) and the vertices are at (0, +12). Since the foci and vertices have the same x-coordinate of 0, we can conclude that the center of the ellipse is also at (0, 0).
The distance from the center to the vertices is given as "a = 12" because the vertices are at (0, +12). Now, let's find the value of "c" using the given information.
Since the foci are at (0, +11), the distance from the center to each focus is given as "c = 11".
Next, we can determine the length of the minor axis by using the relation c^2 = a^2 - b^2, where "b" represents the length of the minor axis.
Plugging in the values, we have:
11^2 = 12^2 - b^2
121 = 144 - b^2
b^2 = 144 - 121
b^2 = 23
Now that we have all the information, we can write the equation of the ellipse with its center at (0, 0) and the given foci and vertices:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
Plugging in the values, we have:
(x - 0)^2 / 12^2 + (y - 0)^2 / 23 = 1
Simplifying:
x^2 / 144 + y^2 / 23 = 1
Therefore, the equation of the ellipse is x^2 / 144 + y^2 / 23 = 1.