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Find the standard form of the equation of the ellipse with the given characteristics.

Foci: (0, 0), (0, 8); major axis of length 18

Please show step by step to answer

the center is midway between the foci, at (0,4)

a = 18/2 = 9
b^2 = 9^2 - 4^2 = 65
so the equation is
x^2/65 + (y-4)^2/81 = 1

Did you know?

Did you know that an ellipse is a shape formed by a curved line that is always equidistant from two fixed points called foci? To find the standard form of the equation of an ellipse, you need the coordinates of the foci and the length of the major axis. In this example, the foci are located at (0, 0) and (0, 8), and the major axis has a length of 18. Step 1: The distance between the two foci gives us the value of 2c, where c is half the distance between the foci. In this case, the distance between (0, 0) and (0, 8) is 8 units, so 2c = 8. Thus, c = 4. Step 2: The length of the major axis, which is the longest line segment that passes through the center of the ellipse, is given as 18 units, so 2a = 18. This means that a = 9. Step 3: The equation of an ellipse in standard form is (x^2 / a^2) + (y^2 / b^2) = 1, where a and b are the lengths of the semi-major and semi-minor axes, respectively. Since a = 9, the squared term in the x coordinate becomes (x^2 / 9^2). We need to find the value of b. Step 4: To find b, we can use the Pythagorean theorem: b^2 = a^2 - c^2. Plugging in the values, we get b^2 = 9^2 - 4^2, which simplifies to b^2 = 81 - 16. Therefore, b^2 = 65. Step 5: Finally, we can rewrite the equation in standard form by substituting the values of a^2 and b^2: (x^2 / 81) + (y^2 / 65) = 1. In summary, the standard form equation of the ellipse with the given foci and major axis length is (x^2 / 81) + (y^2 / 65) = 1.