Equation of the Other Asymptote of the Rectangular Hyperbola
The given equation of the line passing through the center of the rectangular hyperbola is x - y - 1 = 0. One of the asymptotes is 3x - 4y - 6 = 0. To find the equation of the other asymptote, we need to understand the properties of rectangular hyperbolas and how asymptotes are related.

Understanding Rectangular Hyperbolas
- Rectangular hyperbolas have the form xy = c, where c is a constant.
- The center of a rectangular hyperbola is the point where the asymptotes intersect.
- The asymptotes of a rectangular hyperbola pass through the center and are perpendicular to each other.

Finding the Center of the Hyperbola
- The equation x - y - 1 = 0 can be rewritten as y = x - 1, which represents a line passing through the center of the hyperbola.
- By comparing this with the general form xy = c, we can find that the center of the hyperbola is (1, 1).

Relation between Asymptotes and Center
- The equation of one asymptote is given as 3x - 4y - 6 = 0. This line passes through the center of the hyperbola.
- Since the asymptotes are perpendicular, the product of the slopes of the asymptotes is -1.

Finding the Equation of the Other Asymptote
- The slope of the given asymptote 3x - 4y - 6 = 0 is 3/4.
- The slope of the other asymptote will be -4/3 (negative reciprocal of 3/4).
- Using the point-slope form, we can find the equation of the other asymptote passing through the center (1, 1) with a slope of -4/3.
Therefore, the equation of the other asymptote will be of the form y - 1 = -4/3(x - 1), which simplifies to 4x + 3y - 7 = 0.

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