Question

Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$10 x^{2}-7 y^{2}=140$$

Short answer

Question: Given the equation \(10x^2 - 7y^2 = 140\), find the coordinates of the vertices and foci, and the equations of the asymptotes. Answer: The coordinates of the vertices are \((\pm\sqrt{14}, 0)\) and the coordinates of the foci are \((\pm\sqrt{34}, 0)\). The asymptotes have the equations \(y = \pm\sqrt{\frac{10}{7}}x\).

Step-by-step solution

Rewrite the equation in standard form

To rewrite the given equation \(10x^2 - 7y^2 = 140\) in standard form, divide by 140: $$\frac{x^2}{14} - \frac{y^2}{20} = 1$$ Now that we have the hyperbola equation in standard form, we can proceed to find its key features.

Identify the coordinates of the vertices

The equation is in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), which represents a hyperbola with a horizontal transverse axis. The vertices are located at \((\pm a, 0)\). In our equation, \(a^2 = 14\), so \(a = \sqrt{14}\). The vertices are at \((\pm \sqrt{14}, 0)\).

Find the coordinates of the foci

To find the coordinates of the foci, we need to calculate the value of \(c\), where \(c^2 = a^2 + b^2\). In our equation, \(b^2 = 20\), so \(c^2 = 14 + 20 = 34\). Thus, \(c = \sqrt{34}\). The foci are located at \((\pm\sqrt{34}, 0)\).

Find the equations of the asymptotes

The asymptotes have the equations \(y = \pm\frac{b}{a}x\) for hyperbolas with a horizontal transverse axis. In our equation, we have \(a = \sqrt{14}\) and \(b = \sqrt{20}\). The equations of the asymptotes are: $$y = \pm\frac{\sqrt{20}}{\sqrt{14}}x$$ $$y = \pm\sqrt{\frac{20}{14}}x$$ $$y = \pm \sqrt{\frac{10}{7}}x$$

Use a graphing utility to check your work

Now that we have identified the key features, we can use a graphing utility to sketch the graph of the hyperbola with these parameters. Check that the graph matches the given equation, and that the vertices, foci, and asymptotes are correctly identified.

Key concepts

Understanding Vertices of Hyperbolas

Hyperbolas are defined by their unique shape and key features. One of these features is the vertices. For a hyperbola in standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the transverse axis is horizontal. This means the vertices have the coordinates \((\pm a, 0)\). The vertices act as the closest points on each branch of the hyperbola and lie along the transverse axis. For the equation \(\frac{x^2}{14} - \frac{y^2}{20} = 1\), we find that \(a^2 = 14\), giving us \(a = \sqrt{14}\). Therefore, the vertices are positioned at \((\pm \sqrt{14}, 0)\). These points help define the width of the hyperbola's opening.
Understanding vertices is crucial because they provide a frame of reference for other components like the asymptotes and foci.

Exploring Foci of Hyperbolas

Foci are another vital component of hyperbolas. They are located inside each curve of the hyperbola and are used to describe the shape. For hyperbolas like \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the distance \(c\) from the center to each focus is determined by \(c^2 = a^2 + b^2\). This gives a crucial understanding of how the hyperbola stretches.
For our specific hyperbola, \(a^2 = 14\) and \(b^2 = 20\), so \(c^2 = 34\), resulting in \(c = \sqrt{34}\). Thus, the foci are at \((\pm \sqrt{34}, 0)\). These points are essential as they shape the hyperbola's structure by defining its stretching.
The foci highlight how hyperbolas contrast with ellipses, as the foci in hyperbolas exist inside each branch as opposed to being along the same path.

Understanding Asymptotes of Hyperbolas

Asymptotes of a hyperbola are lines that the hyperbola approaches but never touches. They give direction to the curves of the hyperbola as they extend to infinity. For horizontal hyperbolas in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of the asymptotes are \(y = \pm \frac{b}{a}x\).
To find these lines for the equation \(\frac{x^2}{14} - \frac{y^2}{20} = 1\), we calculate \(\frac{b}{a}\) using \(b = \sqrt{20}\) and \(a = \sqrt{14}\). This results in \(y = \pm \sqrt{\frac{10}{7}}x\).
Using these equations, you can visualize how the hyperbola behaves asymptotically, giving it its characteristic shape. Asymptotes provide an understanding of the trajectory and bounds of the hyperbola's arms.

Grasping the Transverse Axis

The transverse axis is the main line that runs through the vertices of a hyperbola and dictates the orientation of its branches. In the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), it is horizontal. This axis is aligned along the x-axis for horizontal hyperbolas. The transverse axis is crucial because it helps define the relation and distance between the vertices and foci.
Knowing the transverse axis aids in determining the direction of the hyperbola and ensures you place key components like the vertices and foci accurately on a graph. It's the central skeleton of the hyperbola's construction, guiding you through its geometry.