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Chapter 9: Problem 69

Describe how to locate the foci of the graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\).

Short Answer

Expert verified
The foci of the graph of the given equation would be located at \((\pm\sqrt{10}, 0)\).

Step by step solution

01

Identify the values of a and b

In the given equation \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\), 'a' is the square root of the number under \(x^{2}\), and 'b' is the square root of the number under \(y^{2}\). Hence, here, \(a=3\) and \(b=1\).
02

Calculate the value of c

Using the formula \(c=\sqrt{a^2+b^2}\), substitute the values of 'a' and 'b' into the formula. This gives \(c=\sqrt{(3)^2+(1)^2}=\sqrt{10}\).
03

Determine the location of the foci

The foci for a horizontal hyperbola are located at \((\pm c, 0)\), hence the coordinates of the foci in this case are \((\pm\sqrt{10}, 0)\).

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Key Concepts

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

Foci of a Hyperbola
The foci of a hyperbola are special points that help capture its unique geometry. Hyperbolas are like inverted curves, and the foci are central to their formation. Understanding focus points is crucial because they are used to describe the hyperbola's shape and orientation.
To find the foci of a hyperbola like \[\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\], follow these steps:
  • Identify the values of \(a\) and \(b\). In a hyperbola equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the denominators tell you \(a^2\) and \(b^2\).
  • Compute the distance \(c\) using \(c = \sqrt{a^2 + b^2}\).
  • For horizontal hyperbolas, the foci are at \((\pm c, 0)\); for vertical, it's \((0, \pm c)\).
In our example, the calculation \(c = \sqrt{10}\) places the foci at \((\pm \sqrt{10}, 0)\). These points show where the hyperbola's branches curve around.
Equation of a Hyperbola
The equation of a hyperbola is a crucial concept when it comes to graphing and understanding this unique curve. Hyperbolas have two standard forms:
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] puts its transverse axis on the x-axis, while
\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\] places it on the y-axis.
In either form, \(a\) and \(b\) come from the denominators, and they determine the shape and size of the hyperbola:
  • \(a\) corresponds to the distance from the center to the vertices along the transverse axis.
  • \(b\) relates to the conjugate axis, influencing the slope of the asymptotes.
For instance, in \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\), \(a=3\) and \(b=1\). These define the vertices and the shape of the curve.
Graphing Hyperbolas
Graphing a hyperbola involves several steps, mainly focusing on understanding its axes and asymptotes. Begin with the hyperbola's equation, such as \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\). From here:
  • Identify \(a\) and \(b\) to set up the vertices and define the axes length.
  • Plot the center at the origin, as it is centered there in this standard form.
  • Determine the vertices, at \((\pm 3, 0)\) in this example, since \(a=3\).
Asymptotes are crucial for the sketch, calculated directly as the lines \(y=\pm \frac{b}{a}x\) or, in our example, \(y=\pm \frac{1}{3}x\). These guide where the curvature approaches infinity.
With these lines and points plotted, you can sketch the two branches of the hyperbola, ensuring they approach the asymptotes but never touch them.

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