Final answer:
To find the standard form of the given ellipse equation, we complete the square for x and y terms and simplify to end up with (x-5)²/25 + (y-4)²/9 = 1. Therefore, the correct answer is Option a.
Step-by-step explanation:
The student is asking to rewrite the equation of an ellipse in standard form. The standard form of an ellipse equation is (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
To convert the given equation 9x² + 25y² - 90x - 200y - 400 = 0 into standard form, we need to complete the square for both x and y terms. First, we group the x terms and the y terms:
9(x² - 10x) + 25(y² - 8y) = 400
Next, we add the squares of half the coefficients of x and y to complete the square:
9[(x - 5)² - 25] + 25[(y - 4)² - 16] = 400
We then simplify the equation by distributing and moving constants to the other side:
9(x - 5)² + 25(y - 4)² - 225 - 400 = 400
9(x - 5)² + 25(y - 4)² = 625
Finally, we divide by 625 to get the standard form:
(x - 5)²/25 + (y - 4)²/9 = 1
Therefore, the correct answer is Option a: (x-5)²/25 + (y-4)²/9 = 1.