91Ó°ÊÓ

The transverse axis is a vital concept in understanding hyperbolas. It's similar to the major axis in an ellipse and indicates the direction along which the hyperbola opens. In this exercise, the hyperbola is defined by the center at \((1, -2)\), vertex at \((3, -2)\), and focus at \((4, -2)\).
The constant y-coordinates and the changing x-coordinates suggest that the transverse axis lies along the x-axis.
For this hyperbola, the transverse axis length is equal to \(2a\), where \(a\) is the distance from the center to a vertex. In this case, we calculate

• \(a = |3 - 1| = 2\)

making the total length of the transverse axis \(2a = 4\).
Understanding this axis helps us know the overall shape and open direction of the hyperbola.

For hyperbolas, the focus is a point that is essential for describing the curve's eccentricity. The distance to the focus, labeled as \(c\), is measured from the center of the hyperbola. In the given problem, the center is at \((1, -2)\) and the focus at \((4, -2)\).
This means the focus lies along the transverse axis (the horizontal direction in this case), and the distance can be easily calculated using the x-coordinates:

• \(c = |4 - 1| = 3\)

The value of \(c\) plays a critical role in forming the hyperbola's standard equation and determining how wide or narrow the hyperbola will be.

The standard form equation of a hyperbola depends on its orientation, center, and lengths associated with \(a\) and \(b\). For a horizontally opening hyperbola, the standard equation is:
\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]
where \((h, k)\) represents the center.
Using \((h, k) = (1, -2)\), \(a^2 = 4\) as calculated from the distance to the vertex, and \(c = 3\), we can apply the formula \(c^2 = a^2 + b^2\) to find \(b^2\):

• \(9 = 4 + b^2\)
• Thus, \(b^2 = 5\)

Inserting all these values gives us the equation:
\[\frac{(x-1)^2}{4} - \frac{(y+2)^2}{5} = 1\]
This equation describes the hyperbola completely in its standard form.