Choose the equation that best represents an ellipse for the given foci and co-vertices.
Foci (+/-3,0)
Co vertices (0, +/-6)
To determine the equation of an ellipse given its foci and co-vertices, we can use the standard form of the equation for an ellipse. This form is:
[(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.
In this case, the given foci are (+/- 3, 0) and the given co-vertices are (0, +/- 6).
Step 1: Find the center (h, k)
The center of the ellipse is found by averaging the coordinates of the foci. Since the foci are symmetrical in the x-axis, the x-coordinate of the center will be the average of the x-coordinates of the foci, while the y-coordinate of the center will be the average of the y-coordinates of the vertices.
Center (h, k) = [(3 + (-3)) / 2, (0 + 0) / 2] = (0, 0)
Step 2: Find the value of "a"
The distance from the center to the vertices is "a". In this case, the distance is the absolute value of the x-coordinate of the foci, which is 3.
a = 3
Step 3: Find the value of "b"
The distance from the center to the co-vertices is "b". In this case, the distance is the absolute value of the y-coordinate of the co-vertices, which is 6.
b = 6
Step 4: Substitute the values into the standard form equation
Substituting the values of (h, k), "a", and "b" into the standard form equation, we get:
[(x - 0)^2 / 3^2] + [(y - 0)^2 / 6^2] = 1
Simplifying further:
x^2/9 + y^2/36 = 1
So, the equation that best represents the ellipse with the given foci and co-vertices is:
x^2/9 + y^2/36 = 1