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Chapter 9: Problem 20

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$9 y^{2}-x^{2}=1$$

Short Answer

Expert verified
The foci of the hyperbola are at positions (0, \(\sqrt{10}\)) and (0, -\(\sqrt{10}\)) while the equations of its asymptotes are \(y = 3x\) and \(y = -3x\).

Step by step solution

01

Identify the form of the equation

The equation is in the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). This indicates a hyperbola with a vertical transverse axis. With the constant on the right side being 1, we can identify that a^2 = 9 and b^2 = 1, implying that a = 3 (the vertical semi-axis length) and b = 1 (the horizontal semi-axis length).
02

Determine the center, vertices, and foci

The center of the hyperbola is at the origin (0,0) because the equation does not involve shifts in the x or y dimension. The vertices are at a distance 'a' above and below the center on the transverse axis. Hence, the vertices are (0, 3) and (0, -3). The distance from the center to the foci is \(\sqrt{a^2 + b^2} = \sqrt{9 + 1} = \sqrt{10}\), so the foci are at (0, \(\sqrt{10}\)) and (0, -\(\sqrt{10}\))
03

Find the equations of the asymptotes

The asymptotes for a hyperbola with a vertical transverse axis have the equations \(y = 卤 \frac{a}{b}x\), here this is \[y = 卤 \frac{3}{1}x\]. So the equations of asymptotes are \(y = 3x\) and \(y = -3x\)
04

Graph the Hyperbola

First graph the center point at (0,0). Then, plot other significant points: the vertices at (0, 3) and (0, -3), and the foci at (0, \(\sqrt{10}\)) and (0, -\(\sqrt{10}\)). Next, draw dashed lines for the asymptotes using the equations \(y = 3x\) and \(y = -3x\). Finally, draw the curve of the hyperbola approaching but never intersecting these asymptotes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertices
In a hyperbola, the vertices are critical points located at the ends of the transverse axis. For the hyperbola given by the equation \(9y^2 - x^2 = 1\), the identified form is \(\frac{y^2}{3^2} - \frac{x^2}{1^2} = 1\). This form indicates a vertical transverse axis.
  • The vertices are at a distance \(a\) from the center along this axis.
  • Since \(a = 3\), the vertices occur at \((0, 3)\) and \((0, -3)\).
These points are where the hyperbola intersects its transverse axis. They are useful when graphing, as they mark the boundaries where each branch of the hyperbola begins.
Asymptotes
Asymptotes are lines that guide the direction of the hyperbola's branches. Although the hyperbola approaches these lines, it never crosses them.
For a vertical hyperbola like \(9y^2 - x^2 = 1\), the asymptotes will intersect at the center of the hyperbola and form diagonals of an imaginary rectangle centered there.
  • The slopes of the asymptotes for a vertical hyperbola are given by \(\pm \frac{a}{b}\).
  • The calculation shows the asymptotes are given by \(y = \pm 3x\).
These asymptotes define the direction each branch of the hyperbola will follow when graphed.
Foci
Along with the vertices, the foci are important components of a hyperbola. They are points that lie along the transverse axis, providing a focal point from which the shape is mathematically defined.
  • For the hyperbola in question, the distance to the foci from the center is found using \(\sqrt{a^2 + b^2}\).
  • Substituting \(a^2 = 9\) and \(b^2 = 1\), the focus distance is \(\sqrt{10}\).
  • Thus, the foci for \(9y^2 - x^2 = 1\) are located at \((0, \sqrt{10})\) and \((0, -\sqrt{10})\).
The presence of the foci helps depict the hyperbola鈥檚 elongation, as they lie outside the vertices on the same line as the transverse axis.
Equations of the Asymptotes
The asymptotic lines of a hyperbola can be calculated from its equation and are essential for understanding its shape. These lines provide a framework for sketching the hyperbola鈥檚 curves accurately.
  • The equations of the asymptotes for \(9y^2 - x^2 = 1\), given the vertical form, use \(y = \pm \frac{a}{b}x\).
  • Here, \(\frac{a}{b} = \frac{3}{1}\), resulting in the equations \(y = 3x\) and \(y = -3x\).
These lines serve as guidelines that the branches of the hyperbola approach, aiding in defining its open-end orientation and providing a basis for sketching the hyperbola accurately.
Graphing a Hyperbola
When graphing a hyperbola, start by plotting its basic components: the center, vertices, and foci. These points anchor the diagram and help convey the hyperbola's overall shape.
  • Begin by marking the center, which for \(9y^2 - x^2 = 1\) is at (0, 0).
  • Plot the vertices at \((0, 3)\) and \((0, -3)\).
  • Next, place the foci at \((0, \sqrt{10})\) and \((0, -\sqrt{10})\).
With these points, draw the asymptotes as dashed lines using \(y = 3x\) and \(y = -3x\). These lines act as guides that the hyperbola branches approach but do not intersect. Finally, sketch the hyperbola鈥檚 curves, showing them approaching the asymptotes.
This process ensures the hyperbola is accurately represented on the coordinate plane.

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