Problem 20 - Precalculus: Mathematics for Calculus

Question

An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. $$y^{2}-25 x^{2}=100$$

Step-by-Step Solution

Answer

Vertices: \((0, 10), (0, -10)\). Foci: \((0, \pm \sqrt{104})\). Asymptotes: \(y = \pm 5x\). Transverse Axis: 20.

Step 1: Identify the Hyperbola Equation Type

The given equation is \( y^2 - 25x^2 = 100 \). It is of the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), which represents a vertical hyperbola since the \( y^2 \) term is positive.

Step 2: Rewrite the Equation in Standard Form

Divide both sides of \( y^2 - 25x^2 = 100 \) by 100 to get it in standard form: \( \frac{y^2}{100} - \frac{x^2}{4} = 1 \). Here, \( a^2 = 100 \) and \( b^2 = 4 \), so \( a = 10 \) and \( b = 2 \).

Step 3: Determine Vertices of the Hyperbola

For a vertical hyperbola, the vertices are located at \((0, \pm a)\). Thus, the vertices are \((0, 10)\) and \((0, -10)\).

Step 4: Calculate the Foci of the Hyperbola

The distance to the foci of a hyperbola is \( c = \sqrt{a^2 + b^2} \). Calculating gives \( c = \sqrt{100 + 4} = \sqrt{104} \). Thus, foci are at \((0, \pm \sqrt{104})\).

Step 5: Find the Asymptotes of the Hyperbola

For a vertical hyperbola, the asymptotes are given by the lines \( y = \pm \frac{a}{b}x \). Substitute \( a = 10 \) and \( b = 2 \), so the asymptotes are \( y = \pm 5x \).

Step 6: Determine the Length of the Transverse Axis

The length of the transverse axis for a vertical hyperbola is \( 2a \). With \( a = 10 \), the transverse axis length is 20.

Step 7: Sketch the Graph of the Hyperbola

To sketch the graph, plot the vertices at \((0, 10)\) and \((0, -10)\). Draw the asymptotes \( y = 5x \) and \( y = -5x \) as dotted lines. Sketch the hyperbola opening vertically away from the center, approaching the asymptotes.

Key Concepts

Vertices

Vertices are key points on a hyperbola where the curve comes closest to its center. In a hyperbola of the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), which represents a vertical hyperbola, the vertices are at the points \((0, \pm a)\).

For our specific hyperbola equation, \(y^2 - 25x^2 = 100\), after simplifying, we identify \(a^2 = 100\), thus \(a = 10\).

• This means the vertices are at \((0, 10)\) and \((0, -10)\).

Vertices are critical for determining the hyperbola's orientation and dimensions.

Asymptotes

Asymptotes of a hyperbola are lines that the curve approaches but never actually touches. They give us an idea of the general shape and spread of the hyperbola. For a vertical hyperbola, the asymptotes are given by the equations:

• \(y = \frac{a}{b}x\)
• \(y = -\frac{a}{b}x\)

In our hyperbola \(\frac{y^2}{100} - \frac{x^2}{4} = 1\), with \(a = 10\) and \(b = 2\), the asymptotes can be calculated as follows:

• The equations \(y = 5x\) and \(y = -5x\) are the asymptotes.

These lines help in sketching the hyperbola by guiding the opening direction and angle of the curves.

Transverse Axis

The transverse axis of a hyperbola is the line that passes through the center and both vertices. It's essentially the "width" of the hyperbola, or the axis along which it opens. For vertical hyperbolas, the transverse axis is the vertical line that includes the two vertices.

In our case:

• The transverse axis is vertical because the hyperbola is vertical.
• Its length is calculated as \(2a\), which for \(a = 10\), gives \(2 \times 10 = 20\).

Understanding the transverse axis is crucial for determining the structure and scale of the hyperbola.

Foci

Just like an ellipse, a hyperbola has two focal points, or "foci", but unlike ellipses, they are located outside the curve. The property of a hyperbola is such that the difference in distances from any point on the hyperbola to the foci is a constant. The distance from the center to each focus can be calculated as \(c = \sqrt{a^2 + b^2}\).

For our equation \(y^2 - 25x^2 = 100\), we find:

• \(c = \sqrt{100 + 4} = \sqrt{104}\)
• Foci are located at \((0, \pm \sqrt{104})\).

This feature is essential for certain applications like satellite navigation, where the hyperbola ensures precise focus placement.

Graphing Hyperbolas

Graphing a hyperbola involves several steps and relies heavily on understanding its main components - vertices, foci, and asymptotes. With a vertical hyperbola, the approach is methodical:

• Firstly, plot the vertices \((0, 10)\) and \((0, -10)\) on the graph.
• Then, draw the asymptotes \(y = 5x\) and \(y = -5x\) as dotted lines; these guide the curvature.
• Next, sketch the curves of the hyperbola, starting at each vertex and extending out, gradually approaching but never crossing the asymptotes.
• Make sure the curve symmetrically opens outwards towards the foci (though not reaching them).

Creating a well-defined graph helps visualize the hyperbola's distinctive shape and understand its unique geometric properties.