Write the equation of an ellipse with the foci (17, 0) and (-17, 0) and co-verticies (0, 15) and (0, -15).
a^2 = b^2 + c^2
a^2 = 225+289 = 514
x^2/514 + y^2/225 = 1
To find the equation of an ellipse given the foci and co-vertices, we can use the general 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 the vertices, and "b" is the distance from the center to the co-vertices.
First, let's find the center of the ellipse:
The center of the ellipse is the midpoint between the two foci.
Center = ( (x₁ + x₂) / 2, (y₁ + y₂) / 2)
Center = ((17 + -17) / 2, (0 + 0) / 2)
Center = (0, 0)
The distance between the center and the vertices, a, is equal to half the distance between the foci.
a = (distance between the foci) / 2
Since the distance between the foci is 2a, a = 17 / 2
a = 8.5
The distance between the center and the co-vertices, b, is equal to half the distance between the co-vertices.
b = (distance between the co-vertices) / 2
Since the distance between the co-vertices is 2b, b = 15 / 2
b = 7.5
Now we have all the necessary values to write the equation of the ellipse:
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
Plug in the values:
(x - 0)^2 / 8.5^2 + (y - 0)^2 / 7.5^2 = 1
Simplify:
x^2 / 8.5^2 + y^2 / 7.5^2 = 1
The equation of the ellipse with the given foci and co-vertices is:
x^2 / 72.25 + y^2 / 56.25 = 1