To find the equation of an ellipse with a major axis length of 10 and foci at (5, 2) and (5, -4), we first need to determine the center of the ellipse. The center of the ellipse is the midpoint between the two foci.
Midpoint formula:
Midpoint_x = (x₁ + x₂) / 2
Midpoint_y = (y₁ + y₂) / 2
For the given foci, we have:
Midpoint_x = (5 + 5) / 2 = 10 / 2 = 5
Midpoint_y = (2 + (-4)) / 2 = -2 / 2 = -1
So, the center of the ellipse is (5, -1).
Next, we find the distance between the two foci, which is equal to the major axis length.
Distance formula:
Distance = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
For the given foci, we have:
Distance = sqrt((5 - 5)² + (-4 - 2)²) = sqrt(0² + (-6)²) = sqrt(0 + 36) = sqrt(36) = 6
The distance between the foci is 6, which represents the length of the major axis. The semi-major axis is half the length of the major axis, so the semi-major axis is 6 / 2 = 3.
Now, we can write the equation of the ellipse using the standard form:
(x - h)² / a² + (y - k)² / b² = 1
Where (h, k) is the center of the ellipse, and 'a' and 'b' represent the semi-major and semi-minor axes, respectively.
Plugging in the values, we have:
(h, k) = (5, -1)
a = 3
Therefore, the equation of the ellipse with a major axis of length 10 and foci at (5, 2) and (5, -4) is:
(x - 5)² / 3² + (y + 1)² / b² = 1
To determine the value of 'b', we can use the Pythagorean theorem for right triangles formed by the foci, the center, and the vertices:
b = sqrt(c² - a²), where c is the distance between the foci
For the given ellipse, c = 6, and a = 3:
b = sqrt(6² - 3²) = sqrt(36 - 9) = sqrt(27) = 3 * sqrt(3)
Therefore, the final equation of the ellipse is:
(x - 5)² / 9 + (y + 1)² / (3 * sqrt(3))² = 1