91Ó°ÊÓ

Hyperbolas are fascinating conic sections distinguished by their characteristic shape. They consist of two separate curves and are defined by subtracting the square of one variable from another. The general equation of a hyperbola is \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] where \((h, k)\) is the center of the hyperbola.
Unlike circles and ellipses, where we see a cohesive curve, hyperbolas are split into two separate branches. Each branch approaches two lines called "asymptotes," but never intersects them. Depending on the positions of \(x\) and \(y\) in the equation, a hyperbola can open either horizontally or vertically. In our solved equation, it opens horizontally because \(x\) is positive.
The beauty and complexity of hyperbolas make them a rich topic in geometry, with applications in navigation, astronomy, and even in architecture.

For hyperbolas, asymptotes are crucial features. They are the invisible lines that the hyperbola approaches but never touches. The equations of these lines provide deep insight into the hyperbola's structure.
The formula for calculating the asymptotes' equations from the standard hyperbola form \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) is given by:
\[ y - k = \pm \frac{b}{a}(x - h) \]
In the given exercise, where both \(a\) and \(b\) are 3, the slope \(\frac{b}{a}\) equals 1. This results in the equations of the asymptotes being \(y = x + 1\) and \(y = -x + 3\).
These straight lines establish the boundaries within which the hyperbola's branches stretch out.

The vertices of a hyperbola are significant points where the curves are closest to each other. These points are on the transverse axis, the line that passes through both the hyperbola's center and its vertices.
To find these vertices, you use the value of \(a\) from the hyperbola's standard form equation.
For our problem, where \(a^2=9\), the value of \(a\) is 3. The vertices are therefore located a distance of 3 units along the x-axis from the center \((1,2)\), resulting in vertices at \((4,2)\) and \((-2,2)\).
These vertices define the extent of the transverse axis and are integral features of the hyperbola's structure. Understanding the location of vertices helps grasp the complete geometry of the hyperbola.

The foci are another set of defining points of a hyperbola. They are located along the transverse axis, outside the vertices. The importance of foci lies in their role in the geometric definition: for any point on the hyperbola, the absolute difference of its distances to the foci is constant.
To find the foci, we calculate \(c\) using the formula \(c^2 = a^2 + b^2\). With our values of \(a^2 = 9\) and \(b^2 = 9\), we have \(c^2 = 18\), giving \(c = 3\sqrt{2}\).
Thus, the foci are positioned at \((1 + 3\sqrt{2}, 2)\) and \((1 - 3\sqrt{2}, 2)\). These points emphasize the open curves of the hyperbola and play a key role in various scientific and engineering calculations.