91Ó°ÊÓ

A hyperbola is a type of conic section—it is a curve that you get when cutting a cone with a plane at a steep angle. Hyperbolas are characterized by their two disjoint curves. If you think of a hyperbola like a pair of mirrored arcs, it may help! The standard form of a hyperbola can be written as \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \). The choice of equation depends on the orientation of the hyperbola, whether it opens horizontally or vertically.

• If the \(x\) term comes first, the hyperbola is horizontal.
• If the \(y\) term comes first, the hyperbola is vertical.

Understanding this basic structure, as in our solution where the equation becomes \( \frac{x^2}{6} - \frac{(y+3)^2}{9} = 1 \), helps you identify a hyperbola and distinguish it from other conic sections, like ellipses.

Vertices are key points on a hyperbola that help define its shape. They are located at the points where the hyperbola intersects with its axis. To find the vertices' positions, we use the equation \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).

To determine the coordinates:

• For a horizontal hyperbola: The vertices lie at \((h \pm a, k)\).
• For a vertical hyperbola: The vertices are at \((h, k \pm b)\).

In the exercise, our hyperbola's vertices at \((\sqrt{6}, -3)\) and \(( -\sqrt{6}, -3)\) tell us it is horizontal because the changes occur along the \(x\)-axis, defined by \(a = \sqrt{6}\).

The foci of a hyperbola are two specific points that lie along the axis of the hyperbola, further from the center than its vertices. They play a crucial role in the geometric definition of a hyperbola. For a hyperbola, the sum of the distances from any point on the hyperbola to the two foci is constant.

The formula to find the foci utilizes \(c\), which is calculated using \(c = \sqrt{a^2 + b^2} \). Once \(c\) is determined:

• For a horizontal hyperbola: The foci are at \((h \pm c, k)\).
• For a vertical hyperbola: The foci are located at \((h, k \pm c)\).

In this problem, calculating \(c\) gives \(c = \sqrt{15}\), placing our foci at \((\sqrt{15}, -3)\) and \((-\sqrt{15}, -3)\), reflecting how they sit outside the vertices along the \(x\)-axis.

Asymptotes are lines that the branches of a hyperbola approach but never touch. They serve as important guides for sketching the hyperbola's shape. Unlike finite parts of the graph, they extend indefinitely.

The equations of the asymptotes for a hyperbola are derived from its standard form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) and are given by:

• \( y = k \pm \frac{b}{a}(x-h) \) for hyperbolas with horizontal branches.
• \( y = k \pm \frac{a}{b}(x-h) \) for hyperbolas with vertical branches.

This equation makes the asymptotes appear as lines cutting through the center of the hyperbola.

In our example, using \( b = 3 \) and \( a = \sqrt{6} \), the asymptotes are \( y = -3 \pm \frac{3}{\sqrt{6}}x \), which simplifies to \( y = -3 \pm \frac{\sqrt{6}}{2}x \). This carefully tells you where the hyperbola's arms spread off to infinity.