Problem 18 - 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. $$25 y^{2}-9 x^{2}=225$$

Step-by-Step Solution

Answer

Vertices: (0, ±3); Foci: (0, ±√34); Asymptotes: y = ±(3/5)x; Transverse axis length: 6.

Step 1: Write the equation in standard form

The given hyperbola equation is \(25y^2 - 9x^2 = 225\). To write it in standard form, divide every term by 225: \[\frac{25y^2}{225} - \frac{9x^2}{225} = 1\]. Simplify to get \(\frac{y^2}{9} - \frac{x^2}{25} = 1\).

Step 2: Identify the hyperbola type and parameters

The standard form \(\frac{y^2}{9} - \frac{x^2}{25} = 1\) indicates a vertical hyperbola. The parameters are \(a^2 = 9\), \(b^2 = 25\), so \(a = 3\) and \(b = 5\). To find \(c\), use \(c^2 = a^2 + b^2\), which gives \(c^2 = 9 + 25 = 34\), hence \(c = \sqrt{34}\).

Step 3: Find the vertices

For a vertical hyperbola, the vertices are at \((0, \pm a)\). Thus, the vertices are \((0, 3)\) and \((0, -3)\).

Step 4: Find the foci

To find the foci, use the formula \((0, \pm c)\) where \(c = \sqrt{34}\). Hence, the foci are at \((0, \sqrt{34})\) and \((0, -\sqrt{34})\).

Step 5: Find the asymptotes

For a vertical hyperbola \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the asymptotes are \(y = \pm \frac{a}{b}x\). Here, \(a = 3\), \(b = 5\), so the asymptotes are \(y = \pm \frac{3}{5}x\).

Step 6: Determine the length of the transverse axis

The transverse axis is vertical with length \(2a\). Given \(a = 3\), the length of the transverse axis is \(2 \times 3 = 6\).

Step 7: Sketch the graph

To sketch the hyperbola, draw the coordinate axes. Plot the vertices at \((0, 3)\) and \((0, -3)\). Sketch the asymptotes \(y = \frac{3}{5}x\) and \(y = -\frac{3}{5}x\). Use these lines as guides to draw the hyperbola's branches, which diverge toward the asymptotes.

Key Concepts

Vertices of Hyperbola

In a hyperbola, vertices serve as key points along the curve where it changes direction. For the hyperbola with the equation \( \frac{y^2}{9} - \frac{x^2}{25} = 1 \), it takes a vertical orientation. This means that its vertices are aligned along the y-axis.
To find the vertices, first identify \( a^2 \) from the standard form, where \( a^2 = 9 \). Thus, \( a = 3 \).
In a vertical hyperbola, the vertices are located at \((0, \pm a)\). Hence, the vertices in this equation are the points \((0, 3)\) and \((0, -3)\).
These vertices not only help in sketching the hyperbola accurately but also signify the closest points between the two branches of the curve.

Foci of Hyperbola

The foci of a hyperbola are essential for understanding the hyperbola's shape and orientation. They lie on the same axis as the vertices but further away from the center.
To find the foci, determine \( c \) using the relationship \( c^2 = a^2 + b^2 \), where \( a^2 = 9 \) and \( b^2 = 25 \). This gives \( c^2 = 9 + 25 = 34 \), so \( c = \sqrt{34} \).
In a vertical hyperbola, the foci are at \((0, \pm c)\), resulting in the points \((0, \sqrt{34})\) and \((0, -\sqrt{34})\).
These foci help in defining the hyperbola's curves as they indicate the direction in which each branch of the hyperbola heads.

Asymptotes of Hyperbola

Asymptotes of a hyperbola are lines that the hyperbola approaches but never touches. They provide a framework, shaping the branches of the hyperbola.
For the vertical hyperbola \( \frac{y^2}{9} - \frac{x^2}{25} = 1 \), the equations of the asymptotes can be given by \( y = \pm \frac{a}{b}x \).
Given \( a = 3 \) and \( b = 5 \), this transformation results in asymptotes as \( y = \frac{3}{5}x \) and \( y = -\frac{3}{5}x \).
These lines act as guides, indicating how the hyperbola curves as \( x \) becomes very large or very small.

Transverse Axis

The transverse axis of a hyperbola is the line segment that connects the two vertices. It lies along the axis of symmetry for the hyperbola.
For vertical hyperbolas like \( \frac{y^2}{9} - \frac{x^2}{25} = 1 \), the transverse axis is vertical and has a length equal to \( 2a \).
With \( a = 3 \), the length is \( 2 \times 3 = 6 \). This axis is crucial as it provides a scale for the hyperbola's overall height and alignment.

Graphing Hyperbola

Graphing a hyperbola involves several steps that align the key components like vertices and asymptotes.
1. **Draw the Coordinate Axes**: Begin with simple horizontal and vertical lines intersecting at the origin.
2. **Plot the Vertices**: Insert marks at points \((0, 3)\) and \((0, -3)\).
3. **Draw Asymptotes**: Draw the lines \( y = \frac{3}{5}x \) and \( y = -\frac{3}{5}x \), extending them across the coordinate grid.
4. **Sketch the Hyperbola**: Based on the vertices and asymptotes, draw the two symmetrical branches of the hyperbola. Each branch approaches, but never crosses, the asymptotes.
This graphical representation helps visualize how the hyperbola behaves and how its structure is influenced by its defining parameters.