Exploring the Equation: (x + 4)(y  5) = 2
The equation (x + 4)(y  5) = 2 represents a hyperbola in the xyplane. Let's break down its characteristics and explore some ways to understand it better.
Understanding the Form

Expanded Form: We can expand the equation to get: xy  5x + 4y  20 = 2
 This form highlights the linear terms (5x and 4y) and the constant term (18).

Hyperbolic Form: The equation is already in a form that directly shows its hyperbolic nature.
 (x + 4)(y  5) = 2 reveals that the product of two linear expressions is a constant. This is a key characteristic of hyperbolas.
Key Features of the Hyperbola

Center: The center of the hyperbola is located at the point (4, 5). This is found by setting both linear expressions equal to zero and solving for x and y.

Asymptotes: The hyperbola has two asymptotes that pass through the center. These lines represent the limiting behavior of the curve as x or y approaches infinity. The equations of the asymptotes can be found by setting each linear expression to zero and solving for y in terms of x, or vice versa:
 y = 5 + (2/x+4)
 x = 4  (2/y5)

Branches: The hyperbola has two branches, one in the first quadrant and one in the third quadrant. This is determined by the positive value on the right side of the equation.
Visualizing the Hyperbola
 Graphing: You can use graphing software or plot points by substituting various values of x and solving for y (or vice versa) to visualize the hyperbola.
 Symmetry: The hyperbola is symmetric about its center. This means that if a point (x, y) lies on the hyperbola, then the point (x  8, y) also lies on the hyperbola.
Applications of Hyperbolas
Hyperbolas have various applications in different fields, including:
 Physics: Describing the trajectories of objects moving under a central force, such as planets orbiting a star.
 Engineering: Designing lenses and mirrors in optical instruments.
 Mathematics: As a fundamental curve in geometry with interesting properties.
Further Exploration
 Transformations: You can explore how transformations like translations and rotations affect the position and orientation of the hyperbola.
 Intersections: You can find points where the hyperbola intersects with other curves or lines by solving systems of equations.
By understanding the key features and properties of the hyperbola represented by the equation (x + 4)(y  5) = 2, you can gain valuable insights into its behavior and applications.