91Ó°ÊÓ

The standard form equation of a hyperbola is essential for understanding its geometric properties. A hyperbola consists of two open curves, and the standard form helps us define their orientation and position in a coordinate plane. There are two general forms of hyperbola equations:

• Horizontal hyperbola: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)
• Vertical hyperbola: \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\)

In these equations, \((h, k)\) represents the center of the hyperbola, while \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. For the given problem, because the hyperbola is aligned horizontally (as the distance between the center and other points is along the x-axis), the horizontal form is used. After plugging in the center \((4, -2)\) and calculated values for \(a\) and \(b\), the standard form becomes:\[\frac{(x-4)^2}{2^2} - \frac{(y+2)^2}{(\sqrt{5})^2} = 1\]

The center of a hyperbola is a key point from which other important elements are measured. This point, denoted by \((h, k)\), serves as the point of symmetry for the hyperbola.

• It determines the midpoint between the two vertices and the midpoint between the foci.
• Its coordinates help us write the hyperbola's equation accurately.

In the exercise, the center is provided as \((4, -2)\). Understanding the center's role allows us to locate the hyperbola correctly on the coordinate plane and apply the standard form equations accordingly. Ensuring accurate interpretation of the center will reflect in how other features, such as vertices and foci, are deduced and calculated.

The focus (plural: foci) of a hyperbola draws significant attention due to its role in the hyperbola's definition and properties. A hyperbola has two foci, and each lies on one of the hyperbola's axes of symmetry. These points are crucial because:

• The difference in distances from any point on the hyperbola to the two foci is constant.
• They are used to derive the hyperbola's defining equation, ensuring its shape and position.

In this exercise, the focus on the horizontal axis is at \((7, -2)\), with the center at \((4, -2)\). The distance \(c\) is calculated as \(|4 - 7| = 3\). The relation \(c^2 = a^2 + b^2\) connects this distance with the other geometric parameters, ensuring the proper shape and placement of the hyperbola in its standard form.

Vertices of a hyperbola are key points directly on the hyperbola, defining the width of its open curves. For a hyperbola:

• Two vertices lie symmetrically on either side of the center.
• The distance from the center to a vertex is defined as \(a\).

In this exercise, given the vertex at \((6, -2)\) and the center at \((4, -2)\), we find that \(a = |4 - 6| = 2\). This calculation is necessary to establish the relationship in the standard form equation \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\). Accurate determination of this distance allows us to confidently derive the equation and understand other related characteristics of the hyperbola.