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The term conic sections refers to the curves that are obtained by intersecting a plane with a double-napped cone. These curves can be identified as parabolas, ellipses, and hyperbolas depending on the angle at which the plane intersects the cone.

For example, when the plane intersects the cone parallel to its side, a parabola is formed. If the plane cuts the cone at a sharp angle but does not go parallel to any side, an ellipse is produced. However, if the plane intersects both naps of the cone, a hyperbola results. These geometric figures are fundamental in various areas of mathematics and science, such as astronomy, physics, and engineering.

The equation given in the exercise, after simplification, represents a hyperbola - a conic section where the plane's intersection angle is steeper than the cone's side, but not perpendicular to the cone's axis, which yields the characteristic two separate curves.

Understanding the vertices and foci of a hyperbola is critical when analyzing its shape and properties. For a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the vertices lie along the x-axis at points \(\pm a, 0\). These points mark the 'tips' of the hyperbola, where it is closest to being 'pointed'.

The foci of a hyperbola, on the other hand, are points from which distances to any point on the hyperbola have a constant difference. These foci are found along the axis of symmetry of the hyperbola, which in this case is the x-axis. The distance to each focus from the center is given by \(c = \sqrt{a^2 + b^2}\), which reflects how 'spread out' the hyperbola is.

In the context of the exercise, we calculated the positions for both the vertices, \(\pm\sqrt{14}, 0\), and the foci, \(\pm\sqrt{34}, 0\), offering a clear illustration of these concepts.

The equations of the asymptotes for a hyperbola provide the slopes of the lines that the hyperbola approaches but never touches—these are the visual 'guidelines' for the open ends of the hyperbola. For the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are determined by \(\frac{b}{a}\), which results in the lines \(\pm\frac{b}{a}y = x\).

In the given exercise, by finding the values of 'a' and 'b' from the hyperbola equation, we can state the asymptote equations as \(\pm\frac{\sqrt{20}}{\sqrt{14}}y = x\). These lines graphically represent the paths the arms of the hyperbola tend toward both in the positive and negative directions indefinitely, thereby outlining the spread of the hyperbola's branches.