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Find an equation the ellipse having a major axis of length 10 and foci at (5,2)and (5,-4)

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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

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