What is the center of the ellipse given by the equation 9x^2 + 25y^2 - 18x + 100y - 116 = 0?
Show the answer and explanation
Correct answer
A. (1, -2)
Principle or equation
To find the center of an ellipse given in general form, complete the square for x and y. The standard form is ((x - h)^2)/a^2 + ((y - k)^2)/b^2 = 1, where (h, k) is the center.
Why this answer is correct
Group x and y terms: 9(x^2 - 2x) + 25(y^2 + 4y) = 116. Complete squares: 9[(x-1)^2 - 1] + 25[(y+2)^2 - 4] = 116. Simplify: 9(x-1)^2 - 9 + 25(y+2)^2 - 100 = 116, so 9(x-1)^2 + 25(y+2)^2 = 225. Divide by 225: (x-1)^2/25 + (y+2)^2/9 = 1. Thus center is (1, -2).
Example
For x^2 + 4y^2 - 4x + 8y + 4 = 0, completing squares gives (x-2)^2/4 + (y+1)^2/1 = 1, center (2, -1).
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