Problem 14 - Algebra and Trigonometry

Question

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. \(x^{2}-2 y^{2}=3\)

Step-by-Step Solution

Answer

Vertices: \((\pm \sqrt{3}, 0)\); Foci: \((\pm \sqrt{4.5}, 0)\); Asymptotes: \(y = \pm \frac{x}{\sqrt{2}}\).

Step 1: Rewrite the Equation in Standard Form

Start by dividing every term of the equation \( x^2 - 2y^2 = 3 \) by 3 to get the equation in standard form. This gives \( \frac{x^2}{3} - \frac{y^2}{1.5} = 1 \).

Step 2: Identify the Components

Compare \( \frac{x^2}{3} - \frac{y^2}{1.5} = 1 \) with the standard form of a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and identify \( a^2 = 3 \), \( b^2 = 1.5 \). Therefore, \( a = \sqrt{3} \) and \( b = \sqrt{1.5} \).

Step 3: Find the Vertices

For a hyperbola in the form \( x^2/a^2 - y^2/b^2 = 1 \), the vertices are at \( (\pm a, 0) \). Thus, the vertices are \((\pm \sqrt{3}, 0)\).

Step 4: Calculate the Foci

The formula for the foci is \( c^2 = a^2 + b^2 \). Calculate \( c \) using \( c = \sqrt{3 + 1.5} \). Therefore, \( c = \sqrt{4.5} \). The foci are at \((\pm \sqrt{4.5}, 0)\).

Step 5: Find the Asymptotes

The asymptotes of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by the equations \( y = \pm \frac{b}{a}x \). Thus, \( y = \pm \frac{\sqrt{1.5}}{\sqrt{3}}x \), which simplifies to \( y = \pm \frac{x}{\sqrt{2}} \).

Step 6: Sketch the Hyperbola

Draw the coordinate axes. Plot the vertices \((\pm \sqrt{3}, 0)\) and foci \((\pm \sqrt{4.5}, 0)\) on the x-axis. Sketch the asymptotes \( y=\pm \frac{1}{\sqrt{2}}x \) as dashed lines, and sketch the hyperbola opening horizontally through the vertices, approaching the asymptotes but not touching them.

Key Concepts

Hyperbola Vertices

The vertices of a hyperbola are important points that lie on the axes on which the hyperbola opens. For the hyperbola described by the equation \( \frac{x^2}{3} - \frac{y^2}{1.5} = 1 \), the vertices are determined by the values of \( a \) and \( b \), which represent the distances from the center to the vertices along the x-axis and y-axis, respectively. In this case, since it is in the horizontal standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the vertices lie on the x-axis. This means the vertices are located at \( (\pm a, 0) \). By identifying \( a = \sqrt{3} \), the vertices are found at \( (\pm \sqrt{3}, 0) \). These points mark the closest and farthest reach of the hyperbola along the x-axis.
Keep in mind that vertices are just the starting points for graphing the hyperbola and give an idea of its width.

Hyperbola Foci

The foci of a hyperbola are specific points located along the major axis that help define the shape and direction of the hyperbola's branches. For the hyperbola in the standard form \( \frac{x^2}{3} - \frac{y^2}{1.5} = 1 \), the foci are found using the relationship \( c^2 = a^2 + b^2 \), where \( c \) is the distance from the center to each focus.
By calculating \( c = \sqrt{3 + 1.5} = \sqrt{4.5} \), we find the foci at points \( (\pm \sqrt{4.5}, 0) \). These points are positioned further out from the vertices along the x-axis and help locate the path that each branch of the hyperbola will follow.
The understanding of foci is crucial because they are used in applications like orbital paths in physics and engineering, showcasing the importance of accurate placement in graphing.

Hyperbola Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches. They give a clear indication of how wide or narrow the hyperbola is and serve as a guide for sketching the curve.
For a hyperbola in this form, the asymptotes are determined using the equation \( y = \pm \frac{b}{a}x \). In our case, \( y = \pm \frac{\sqrt{1.5}}{\sqrt{3}}x \), which simplifies to \( y = \pm \frac{x}{\sqrt{2}} \). These lines cross through the origin and have slopes that depend on \( b \) and \( a \), and are integral to the hyperbola's definition.
Visualizing asymptotes helps understand how the branches of the hyperbola open and extend infinitely but never cross these guidelines, making them essential for a precise graph.

Standard Form of Hyperbola

The standard form of a hyperbola is pivotal in understanding its geometric properties and graphing it accurately. This form is generally expressed as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) for a horizontally oriented hyperbola, or as \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) for a vertical orientation.
For the given hyperbola, we started with the equation \( x^2 - 2y^2 = 3 \). By dividing every term by 3, we transformed it to \( \frac{x^2}{3} - \frac{y^2}{1.5} = 1 \). This transformation is crucial to identify the parameters \( a^2 = 3 \) and \( b^2 = 1.5 \), where \( a = \sqrt{3} \) and \( b = \sqrt{1.5} \).
Knowing the standard form allows us to quickly find properties such as vertices, foci, and asymptotes, making graphing and further analysis straightforward and consistent.