91Ó°ÊÓ

The vertices of a hyperbola are the points where the hyperbola makes its closest approach to its center. In this exercise, the given vertices are \((5, -2)\) and \((1, -2)\), which lie on the same horizontal line. The distance between these two points determines the length of the transverse axis, which is 4 units in this case since the difference in the x-coordinates is 4. By definition, the midpoint of these vertices serves as the center of the hyperbola, which we calculated to be \((3, -2)\).

This center is crucial because it helps us shift the hyperbola along the coordinate plane to fit its standard equation. Understanding the location of the vertices thus not only identifies the stretch of the hyperbola but also its orientation, whether the transverse axis is horizontal or vertical.

Asymptotes are the lines that a hyperbola approaches but never touches. They give us an essential insight into the hyperbola's shape and orientation. For hyperbolas, asymptotes intersect at the center and define guiding lines that extend diagonally. In this problem, the asymptotes are given by the equations \(y=\pm \frac{3}{2}(x-3)-2\).

The slopes of these asymptotes, \(\pm \frac{3}{2}\), indicate the steepness and direction of the lines. With the center at \((3, -2)\), these asymptotes help determine the relative width of the hyperbola's branches. For a hyperbola of the form we are working with, the slope relates to the values \(a\) and \(b\) as follows: \(\pm \frac{b}{a}\). This understanding allows us to solve for the parameter \(b\) in the hyperbola's equation once \(a\) is known.

The transverse axis of a hyperbola is the axis along which the vertices lie. It defines the primary direction of the hyperbola's opening. In this exercise, because the vertices \((5, -2)\) and \((1, -2)\) share the same y-coordinate, the transverse axis is horizontal. This contributes to the choice of equation form for the hyperbola:

• \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) when horizontal
• \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\) when vertical

For this hyperbola, the transverse axis length \(2a\) is determined by the distance between the vertices, measured at 4 units here, leading to \(a = 2\).

Understanding whether the axis is transverse or conjugate helps determine how to interpret the dimensions \(2a\) (distance between vertices) and \(2b\) (distance between asymptotes' crossing paths).

The equation of a hyperbola reveals its complete geometric representation. Based on this exercise, knowing the vertices and asymptotes allows us to derive the equation. With the center at \((3, -2)\), \(a = 2\), and \(b = 3\), the hyperbola's equation is in the standard form:

\[\frac{(x-3)^2}{4} - \frac{(y+2)^2}{9} = 1\]

This equation shows that around the center, the square of the x-axis component divided by \(a^2\) minus the y-axis component divided by \(b^2\) equals 1. The hyperbola will open in the direction of the transverse axis, which is horizontal in this example.

• \((x-h)^2\) and \((y-k)^2\) shift the center of the hyperbola from the origin to \((h, k)\)
• \(a\) determines the distance from the center to each vertex along the transverse axis
• \(b\) affects how much the hyperbola stretches in the perpendicular direction

This equation encapsulates the hyperbola's shape and position relative to the coordinate plane, giving a mathematical model for all possible points \((x, y)\) that lie on the hyperbola.