Problem 89 - Algebra and Trigonometry

Question

What happens to the shape of the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) as \(\frac{c}{a} \rightarrow \infty,\) where \(c^{2}=a^{2}+b^{2} ?\)

Step-by-Step Solution

Answer

As \(c/a\) tends towards infinity, the hyperbola flattens becoming a pair of vertical lines along the lines \(x = ±a\).

Step 1: Derive Expression for c/a

Given that \(c^2 = a^2 + b^2\), we can rearrange terms and get \(c/a = \sqrt{1 + (b/a)^2}\). This relation shows how \(c/a\) is affected by the change in \(b/a\). Now, what happens when \(c/a\) approaches infinity?

Step 2: Analyze Overshoot

As \(c/a\) approaches infinity, it implies that \(a\) is getting smaller while \(c\) stays constant or \(c\) is getting larger while \(a\) stays constant, or both. This means that the term \(b/a\) in the above relation becomes very large, and hence \(b\) is much greater than \(a\). Thus, the hyperbola is becoming more elongated along the y-axis.

Step 3: Final Conclusion

As \(c/a\) tends towards infinity, the hyperbola becomes a pair of vertical lines along the lines \(x = ±a\). This is because the distance between the center and the foci (c) increases to a large extent compared to the semimajor axis (a). Thus the foci tends to be like a straight vertical line as \(c/a\) tends towards infinity.

Key Concepts

Conic Sections

Conic sections are the various shapes created when a plane intersects a cone. Depending on the angle and position of the intersection, we can get different figures such as circles, ellipses, parabolas, and hyperbolas. Each shape has a unique set of properties and equations that describe it. For instance, a hyperbola is formed when the plane intersects both nappes of the cone and the angle of intersection is smaller than that of the cone's axis. It is important to understand that conic sections are not just mathematical concepts, but they have practical applications in fields such as astronomy, physics, and engineering.

Hyperbola Asymptotes

Hyperbolas are characterized by their asymptotes, which are the lines that a hyperbola approaches but never touches. These asymptotes provide a sort of 'boundary' to the shape of a hyperbola. In the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), the asymptotes would be the lines defined by \(y = \pm\frac{b}{a}x\). Exploring what happens as \(\frac{c}{a} \rightarrow \infty\), which is tied to the foci of the hyperbola, gives insight into the behavior of these asymptotes. In such a case, the hyperbola stretches along the y-axis and the asymptotes become nearly vertical. This behavior can be visualized and better understood with interactive graphing tools or software that allow manipulation of the hyperbola's parameters.

Semimajor and Semiminor Axes

The semimajor and semiminor axes are key to understanding the geometry of a hyperbola. The semimajor axis, denoted by \(a\), is the distance from the center to the farthest point on the hyperbola along the axis of symmetry. The semiminor axis, represented as \(b\), is the distance from the center to the closest point on the curve. These two axes are perpendicular to each other, and they intersect at the hyperbola's center. When analyzing the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and considering the limit as \(\frac{c}{a} \rightarrow \infty\), the size of the semiminor axis \(b\) becomes larger relative to \(a\), showing the hyperbola's increasing elongation.

Hyperbola Equations

The general equation of a hyperbola centered at the origin with horizontal transverse axis is \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) and that with a vertical transverse axis is \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\). These equations provide a precise representation of the hyperbola's position and shape. By plugging in values for \(x\) and \(y\), we can find the corresponding points on the hyperbola. It's crucial to note that the constant terms \(a\) and \(b\) are squared in the equations, signifying their roles as the squares of the distances of the semimajor and semiminor axes respectively. When \(c/a\) approaches infinity, as in the exercise, this affects the shape of the hyperbola so profoundly that it essentially becomes a completely different figure: a pair of vertical lines.