Write and equation of the ellipse with foci at (0,土11) and vertices at (0,土12)
AAAaannndd the bot gets it wrong yet again!
Clearly, the center is at (0,0), and we have
b^2 = a^2-c^2 = 12^2 - 11^2 = 144-121 = 23
and the major axis is vertical, so we have
x^2/23 + y^2/144 = 1
I apologize for the mistake in my previous response. You are correct that the center is at (0, 0) and the major axis is vertical. The correct equation of the ellipse is:
x^2/23 + y^2/144 = 1
To find the equation of an ellipse with the given information, we can utilize the standard form for the equation of an ellipse centered at the origin:
(x^2/a^2) + (y^2/b^2) = 1
Here, a represents the distance from the center to a vertex, and c represents the distance from the center to a focus. To find the values of a and c, we can make use of the given information:
The vertices are located at (0, √12), which means a = √12.
The foci are located at (0, √11), which means c = √11.
Now we can substitute these values into the equation:
(x^2/(√12)^2) + (y^2/b^2) = 1
Simplifying:
(x^2/12) + (y^2/b^2) = 1
Therefore, the equation of the ellipse is:
(x^2/12) + (y^2/b^2) = 1
The standard form of the equation of an ellipse with foci at (0, b) and vertices at (0, a) is:
((x - 0)^2) / ((a^2) - (b^2)) + ((y - b)^2) / (a^2) = 1
Plugging in the given values, we get:
((x - 0)^2) / ((12^2) - (11^2)) + ((y - 11)^2) / (12^2) = 1
Simplifying further:
(x^2) / 23 + ((y - 11)^2) / 144 = 1
Therefore, the equation of the ellipse is:
(x^2) / 23 + ((y - 11)^2) / 144 = 1