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The hyperbola is one of the four types of conic sections—shapes formed by the intersection of a plane and a double-napped cone. The standard form of a hyperbola that opens horizontally is given by the equation:

• \[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]

Here,

• \((h, k)\) is the center of the hyperbola,
• \(a\) is the distance from the center to each vertex along the line of symmetry,
• \(b\) is the distance related to the slope of the asymptotes.

To distinguish a hyperbola from other conic sections, notice the minus sign between the squares of \(x\) and \(y\). In a hyperbola that opens horizontally, the \(x\)-terms have a positive fraction, while the \(y\)-terms have a negative fraction. This shape creates two branches that look like mirror images of each other. They extend indefinitely, approaching but never touching their asymptotes. The asymptotes intersect at the center \((h, k)\), guiding the curvature of the hyperbola.

Completing the square is a useful algebraic technique for transforming a quadratic expression into a perfect square trinomial. This method helps in finding the standard form of conic sections such as ellipses, parabolas, and hyperbolas. Here’s how it works:

• Given a quadratic expression, \(ax^2 + bx + c\), focus first on \(x\)-related terms.
• Rearrange and factor out the leading coefficient from \(x^2\) and \(x\) terms if necessary. For example, \(16(x^2 - 6x)\).
• Take the coefficient of \(x\), divide by two, and square it. For instance, divide \(-6\) by \(2\) to get \(-3\), square it to obtain \(9\).
• Add and subtract this square inside the expression, effectively maintaining the equality: \((x^2 - 6x + 9 - 9)\).
• Simplify to reveal a perfect square trinomial: \((x-3)^2\).

By completing the square in the original exercise, we reformat the problem into a recognizable form, allowing us to identify it as a hyperbola. This technique is invaluable for shifting from general quadratic forms to easily interpreted conic section equations.

An ellipse is another type of conic section, resembling an elongated circle. Its standard form equation is:

• \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]

Unlike the hyperbola, the ellipse has plus signs between the squared terms. Here,

• \((h, k)\) is the center of the ellipse,
• \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

In general,

• If \(a > b\), the ellipse is stretched more along the \(x\)-axis.
• If \(b > a\), it is stretched along the \(y\)-axis.

Ellipses differ from hyperbolas as they are closed shapes and do not have asymptotes. They encapsulate a finite area, making them fitting models for planetary orbits and other similar natural phenomena.

Vertices and foci are critical components when analyzing conic sections, specifically for ellipses and hyperbolas.

• **Vertices**: These are the points where the conic section intersects its major axis. In a hyperbola, they lie on the transverse axis—the line segment that passes through the center and vertices. The equation \((h \pm a, k)\) gives the vertices of a horizontally opening hyperbola and \((h, k \pm b)\) for a vertically opening one.
• **Foci**: These are specific points from which distances to any point on the hyperbola maintain a constant difference. For a hyperbola centered at \((h, k)\), the foci are located at \((h \pm c, k)\) where \(c\) is calculated using \(c^2 = a^2 + b^2\).

• In ellipses, the vertices are also determined by the distances along the major axis, which are \((h \pm a, k)\) for horizontal or \((h, k \pm b)\) for vertical axes.
• The foci of an ellipse lie on the major axis too, governed by \(c^2 = a^2 - b^2\).

Understanding these geometrical elements helps in sketching and interpreting the shapes, thus making conic sections more comprehensible.