Problem 17 - Algebra and Trigonometry

Question

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ x^{2}-4 y^{2}-8=0 $$

Step-by-Step Solution

Answer

Vertices are at \( (\pm 2\sqrt{2}, 0) \); foci at \( (\pm \sqrt{10}, 0) \); asymptotes are \( y = \pm \frac{1}{2}x \).

Step 1: Rewrite the Equation in Standard Form

The given equation is \( x^2 - 4y^2 - 8 = 0 \). To rewrite this in standard form for a hyperbola, we first add 8 to both sides to get \( x^2 - 4y^2 = 8 \). Then, we divide every term by 8 to attain \( \frac{x^2}{8} - \frac{y^2}{2} = 1 \). This is now in the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

Step 2: Identify the Center, a, and b

From the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), we can identify that \( a^2 = 8 \) and \( b^2 = 2 \). Therefore, \( a = \sqrt{8} = 2\sqrt{2} \) and \( b = \sqrt{2} \). The center of a hyperbola in this form is at the origin, (0,0).

Step 3: Find the Vertices

The vertices of the hyperbola are located at \( (\pm a, 0) \) for this orientation. So, substituting \( a = 2\sqrt{2} \), we get the vertices at \( (2\sqrt{2}, 0) \) and \( (-2\sqrt{2}, 0) \).

Step 4: Calculate the Foci

To find the foci, we use the formula \( c^2 = a^2 + b^2 \). Substituting \( a^2 = 8 \) and \( b^2 = 2 \), we find \( c^2 = 8 + 2 = 10 \), giving \( c = \sqrt{10} \). Thus, the foci are located at \( (\pm \sqrt{10}, 0) \).

Step 5: Determine the Asymptotes

The asymptotes of the hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by the equations \( y = \pm \frac{b}{a}x \). Substituting \( a = 2\sqrt{2} \) and \( b = \sqrt{2} \), we find the asymptote equations: \( y = \pm \frac{\sqrt{2}}{2\sqrt{2}}x = \pm \frac{1}{2}x \).

Step 6: Sketch the Hyperbola

Drawing the hyperbola involves plotting the center at (0,0), plotting vertices at (\(2\sqrt{2} \),0) and (\(-2\sqrt{2} \),0), the foci at (\(\sqrt{10}\),0) and (\(-\sqrt{10}\),0), and sketching the asymptotes \( y = \frac{1}{2}x \) and \( y = -\frac{1}{2}x \). The hyperbola will open left and right along the x-axis, approaching these asymptotes.

Key Concepts

Vertices of a Hyperbola

The vertices of a hyperbola are key points that help you understand the shape and orientation of the curve. In the standard form equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the vertices are located along the axis parallel to which the hyperbola opens. Since the hyperbola in question opens along the horizontal axis, the vertices are found at \((\pm a, 0)\).
For our equation, we determined \( a = 2\sqrt{2} \). This puts the vertices at \((2\sqrt{2}, 0)\) and \((-2\sqrt{2}, 0)\). These are the points where the hyperbola gets closest to the center, located at the origin (0,0). Knowing the vertices allows you to sketch the initial shape of the hyperbola, showing where it stretches along its main axis. The line segment connecting the vertices is the "transverse axis," which is also a central line of symmetry for the hyperbola.

Foci of a Hyperbola

The foci of a hyperbola are points from which the distances help shape the hyperbola's curves. The positioning of the foci determines how wide or narrow the hyperbola appears. For the equation in standard form, \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the foci are located at \((\pm c, 0)\), where \( c \) is found using the relationship \( c^2 = a^2 + b^2 \).
In our specific example, we have \( a^2 = 8 \) and \( b^2 = 2 \), leading to \( c^2 = 10 \). Calculating further, \( c = \sqrt{10} \). Thus, the foci are positioned at \((\sqrt{10}, 0)\) and \((-\sqrt{10}, 0)\). The foci help guide the drawing of the hyperbola by providing an important geometrical aspect that affects the curve's "spread." Always remember that these points remain inside the arms of the hyperbola.

Asymptotes of a Hyperbola

Asymptotes of a hyperbola are invisible lines that the curve approaches but never intersects. They essentially guide the "sheath" of the hyperbola, showing the direction it opens in. For our specific form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes can be calculated using the equation \( y = \pm \frac{b}{a}x \).
Given that \( a = 2\sqrt{2} \) and \( b = \sqrt{2} \), the asymptote equations simplify to \( y = \pm \frac{1}{2}x \). These lines pass through the origin (0,0). Plotting these will form an "X" across the center and show the general shape's borders. Asymptotes are crucial for accurately sketching a hyperbola, giving insight into where the curve will extend in both directions. Remember that a hyperbola never actually reaches or crosses these asymptotes; they simply suggest the trend of the arms as they extend indefinitely in space.