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The equation x² - 4y² - 2x - 40y - 103 = 0 can be expressed as an ellipse in standard form by completing the square for both x and y terms.
First, rearrange the equation by grouping the x and x terms together and the y and y terms together:
(x² - 2x) - 4(y² + 10y) = 103
Complete the square for x and y terms:
[(x - 1)² - 1] - 4[(y + 5)² - 25] = 103
Expand the squared terms:
(x - 1)² - 4(y + 5)² + 100 = 103
Rearrange and simplify:
(x - 1)² - 4(y + 5)² = 3
Therefore, the equation of the ellipse in standard form is:
(x - 1)²/3 - 4(y + 5)²/3 = 1
This standard form allows for a clear interpretation of the ellipse's center, semi-major and semi-minor axes lengths, and orientation.