What is an equation for the ellipse with foci (0, -5) and (0,5) and vertices (0, -9) and (0, 9)
The center of the ellipse is at the midpoint of the foci, which is (0,0). The distance between the center and a vertex is the length of the semi-major axis a, which is 9. The distance between the center and a focus is c, which is 5. The equation for the ellipse is:
(x-0)^2 / 9^2 + (y-0)^2 / b^2 = 1
where b is the length of the semi-minor axis. To find b, we use the relationship:
b^2 = a^2 - c^2
b^2 = 9^2 - 5^2
b^2 = 56
Therefore, the equation for the ellipse is:
x^2 / 81 + y^2 / 56 = 1
AAAaannndd the bot gets it wrong yet again!
and after all the correct work, your final answer is wrong.
wrong, since the major axis is vertical. The correct equation is
x^2/56 + y^2/81 = 1
I apologize for my mistake. You are correct, the major axis is vertical and the correct equation for the ellipse is:
x^2/56 + y^2/81 = 1
Thank you for correcting me.
To find the equation for the ellipse with the given foci and vertices, we can use the standard equation for an ellipse:
[(x - h)^2 / a^2] + [(y - k)^2 / b^2] = 1
where (h, k) represents the center of the ellipse, a is the distance from the center to each vertex along the x-axis, and b is the distance from the center to each vertex along the y-axis.
In this case, we can observe that the center of the ellipse is at the origin (0, 0).
The distance between the center and each vertex along the x-axis is 9, so a = 9.
The distance between the center and each vertex along the y-axis is 5, so b = 5.
Now, we need to find the value of c, which is the distance from the center to each focus.
From the given foci, we can calculate the value of c by using the distance formula:
c = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
Applying this formula:
c = sqrt[(0 - 0)^2 + (5 - (-5))^2]
= sqrt[0 + 10^2]
= sqrt(100)
= 10
Now that we have the values of a, b, and c, we can substitute them into the standard equation to obtain the equation for the ellipse:
[(x - 0)^2 / 9^2] + [(y - 0)^2 / 5^2] = 1
Simplifying it further:
[x^2 / 9^2] + [y^2 / 5^2] = 1
The equation for the ellipse is:
x^2 / 81 + y^2 / 25 = 1