Question

Write an equation for the given ellipse that satisfies the following conditions. Center at (5,5); minor axis vertical, with length 16;c=6. The equation for the given ellipse is (Type your answer in standard form.)

Answer 1

The equation for an ellipse with a center at (5,5), a vertical minor axis of length 16, and a distance between foci of 6 is given by ((x - 5)² / 100) + ((y - 5)² / 64) = 1 in standard form.

Step-by-step explanation:

To write the equation for an ellipse with a center at (5,5), a vertical minor axis with length 16, and a distance between foci (c) of 6, we must first determine the lengths of the semi-major axis (a) and the semi-minor axis (b). The given minor axis length is 16, so the semi-minor axis is b = 8 (half the minor axis).

The distance between the foci is 2c, which means each focus is c = 6 away from the center. The relationship between a, b, and c is given by the equation c² = a² - b².

Plugging in the values we have, we solve for a:

6² = a² - 8²
36 = a² - 64
a² = 100
a = 10

Now, the standard form of the equation of an ellipse centered at (h,k) with a vertical minor axis is:

((x - h)² / a²) + ((y - k)² / b²) = 1

Substituting our values we get:

((x - 5)² / 10²) + ((y - 5)² / 8²) = 1

Therefore, the equation for our ellipse in standard form is:

((x - 5)² / 100) + ((y - 5)² / 64) = 1