91Ó°ÊÓ

In a hyperbola, the semi-major axis, denoted as \(a\), is the longest radius that extends from the center to the vertex along the transverse axis of the hyperbola. For a hyperbola centered at the origin, this is the distance from \( (0,0) \) to \( (a, 0) \) along the x-axis. This distance is one of the principal components that help define the shape and size of the hyperbola.

It’s important to remember that in a hyperbola, unlike in an ellipse where \(a\) represents the longest axis, the length of the transverse axis of the hyperbola dictates the spread of the hyperbola branches. For example, in the given exercise, \(a = 3\), implying the distance from the center to each vertex on the transverse axis is 3 units. Understanding the semi-major axis is crucial for accurately solving any problem related to hyperbolas because it directly interacts with the other key parameters of the hyperbola.

The semi-minor axis, denoted as \((b)\), is the shortest radius extending from the center to the co-vertices along the conjugate axis of the hyperbola. For a hyperbola, this axis passes through the center and is perpendicular to the transverse axis.

Using the relation \((c^2 = a^2 + b^2)\), we can find the value of \((b)\) when \((a)\) and \((c)\) are known. In our exercise, we know that \((a = 3)\) and \((c = 5)\). Plugging these values into the equation gives us:
\[ 5^2 = 3^2 + b^2 \]
Simplifying, we find:
\[ 25 = 9 + b^2 \]
\[ 25 - 9 = b^2 \]
\[ 16 = b^2 \]
Taking the square root of both sides, we get \((b = 4)\). This means the semi-minor axis of this hyperbola is 4 units, which completes the essential parameter set for our hyperbola.

Another key aspect of a hyperbola is the distance from the center to the foci, denoted as \((c)\). The foci are the two points located along the transverse axis, equidistant from the center, and play a fundamental role in defining the hyperbola.

For hyperbolas, the relationship between \(a\), \(b\), and \(c\) is given by \(( c^2 = a^2 + b^2 )\). These foci points are essential as they help in understanding many properties of the hyperbola, such as the equation of its asymptotes.

In the given exercise, \(c = 5\), meaning the distance from the center \( (0,0) \) to each focus is 5 units. This reveals that the focal points are positioned at \((\textpm 5, 0)\) along the transverse axis. Hence, knowing the distance to the foci helps better visualize and construct the graph of the hyperbola.