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When dealing with hyperbolas, vertices are crucial points that help us identify the shape and orientation. In the context of the given problem, the vertices of the hyperbola are

• \((2, 3)\), and
• \((2, -3)\).

These points are symmetrically located above and below the center. The center, found horizontally between the vertices, is at

• \((2, 0)\).

Understanding the vertices helps determine how stretched or compressed the hyperbola is vertically.
The distance from the center to each vertex, known as half the transverse axis length, is represented by \(a\). In this case, \(a = 3\). This tells us the hyperbola's shape is more extended along the vertical direction.

Foci are special points inside the hyperbola that define the curve's unique property: the difference of distances from any point on the hyperbola to the two foci is constant.
For our hyperbola, the foci are located at

• \((2, 5)\), and
• \((2, -5)\).

These lie further from the center than the vertices, indicating the focus of a hyperbola.
The distance from the center to a focus is represented by \(c\). Here, \(c = 5\). This distance plays a crucial role in define the hyperbola, as it helps us calculate \(b\) - the half-length of the conjugate axis using the relationship \(c^2 = a^2 + b^2\).
With \(a = 3\) and \(c = 5\), we find \(b = 4\). This interrelationship of \(a\), \(b\), and \(c\) defines the hyperbola's precise geometry.

A hyperbola possesses several distinct properties that differentiate it from other conic sections like ellipses and parabolas. One major property is that it consists of two disconnected curves called branches.
For the given equation of the hyperbola \[\frac{y^2}{9} - \frac{(x-2)^2}{16} = 1,\]we see that this is a vertical hyperbola because the \(y\) term comes first.
Some key properties include:

• **Asymptotes:** These are the lines that the hyperbola approaches but never touches. For a vertical hyperbola like ours, the equations of the asymptotes are derived from its equation.
• **Center:** Located at \((h, k)\) which is \((2, 0)\) in our case.
• **Axes:** The transverse axis is vertical with length \(2a\) and the conjugate axis is horizontal with length \(2b\).

Understanding these properties helps visualize the hyperbola, ensuring you can graph it accurately and appreciate its unique characteristics in coordinate space.