Determine the new equation of xy - x - y - 10 = 0 if the origin is translated to the point (1, 1).
To determine the new equation of the given equation xy - x - y - 10 = 0 after translating the origin to the point (1, 1), we need to perform a coordinate transformation.
Let's denote the original coordinates as (x, y) and the new coordinates as (x', y'). The translation of the origin by (1, 1) can be represented by the equations:
x' = x - 1
y' = y - 1
Substituting these new coordinates into the original equation, we get:
(x' + 1)(y' + 1) - (x' + 1) - (y' + 1) - 10 = 0
Expanding the parentheses, we have:
(x'y' + x' + y' + 1) - (x' + 1) - (y' + 1) - 10 = 0
Combining like terms, we simplify to:
x'y' + x' + y' - x' - y' - 9 = 0
Further simplifying, we have:
x'y' - 9 = 0
Thus, the new equation of xy - x - y - 10 = 0 after translating the origin to the point (1, 1) is x'y' - 9 = 0.