Question

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes, and directrices. Use a graphing utility to check your work. A hyperbola with vertices (±1,0) and eccentricity 3

Short answer

Answer: The equation of the hyperbola is \(x^2 - \frac{y^2}{8} = 1\). The vertices are at (±1, 0), the foci are at (±3, 0), the asymptotes are given by the lines y = ±(\sqrt{8}/1) * x, and the directrices are the vertical lines x = ±(1/3).

Step-by-step solution

Determine the values of a and c for the hyperbola equation

The equation of a hyperbola centered at the origin has the following standard form: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] We are given the vertices (±1,0). Since they lie on the x-axis, we know that a = 1 (the distance from the center to the vertex). The relationship between the eccentricity (e), a, and c for a hyperbola is: \[e = \frac{c}{a}\] Given eccentricity (e) = 3, we can find the value of c: \[c = e \cdot a\] \[c = 3 \cdot 1\] \[c = 3\]

Find the value of b using the relationship between a, b, and c

The relationship between a, b, and c for a hyperbola is given by: \[c^2 = a^2 + b^2\] We already have the values for a and c from the previous step (a = 1, c = 3). So we can find the value of b: \[b^2 = c^2 - a^2\] \[b^2 = 9 - 1\] \[b^2 = 8\] \[b = \sqrt{8}\]

Find the equation of the hyperbola

Now that we have the values for a and b, we can find the equation of the hyperbola: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] \[\frac{x^2}{1^2} - \frac{y^2}{(\sqrt{8})^2} = 1\] \[\frac{x^2}{1} - \frac{y^2}{8} = 1\] So the equation of the hyperbola is: \[x^2 - \frac{y^2}{8} = 1\]

Sketch the graph of the hyperbola and label the vertices, foci, asymptotes, and directrices

Vertices: (±1,0) Foci: (±3,0) Asymptotes: y = ±(\sqrt{8}/1) * x Directrices: vertical lines at x = ±(a/e) Directrices: x = ±(1/3) Plot these points and lines on the graph to visualize the hyperbola, making sure to label each point and line. After completing the sketch, verify your work using a graphing utility.

Key concepts

Eccentricity

In the world of conic sections, eccentricity offers a way to describe how elongated a hyperbola is. For a hyperbola, the eccentricity (\( e \)) is always greater than one. This crucial characteristic distinguishes it from ellipses, which have eccentricities less than one.
To better appreciate this concept, recall the formula that relates eccentricity, the distance to the foci (\( c \)), and the distance to the vertices (\( a \)):

• \( e = \frac{c}{a} \)

In the given problem, the vertices of the hyperbola are (\( \pm 1,0 \)), indicating that \( a \) is 1. With the eccentricity provided as 3, we can compute \( c \) by multiplying the eccentricity by \( a \, \), resulting in \( c = 3 \).
Understanding eccentricity helps in determining how 'spread out' the branches of the hyperbola are. As the eccentricity increases, the branches of the hyperbola become more open, emphasizing its elongated characteristic.

Vertices of Hyperbolas

The vertices of a hyperbola are significant points where the curve reaches its closest distance to the center. For a hyperbola centered at the origin, these vertices align along the transverse axis, which can either be horizontal or vertical. This axis determines if the hyperbola opens left and right, or up and down.
In the exercise, the vertices are located at (\( \pm 1, 0 \)), defining a horizontal transverse axis. This placement implies the hyperbola opens towards the sides rather than up and down.

• Transverse Axis: the line joining the vertices, also dictating the direction of the hyperbola’s open branches.
• Formula: distance from the center to each vertex is given by \( a \).

The knowledge of vertices assists in drafting the structure of the hyperbola and provides a reference for constructing the hyperbola's equation. With vertices specified, it’s straightforward to identify the transverse axis and further properties like asymptotes and foci.

Properties of Hyperbolas

Hyperbolas showcase a unique set of features that help to identify and analyze them. One of those properties is that they consist of two separate branches. Each branch resembles a `U` shape but with precise mathematical curvature.
Let’s explore crucial properties of hyperbolas:

• Equation: The standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or vice versa depending on the orientation.
• Asymptotes: Lines that the hyperbola approaches but never intersects. They provide a framework shaping the trajectory of the branches and follow the equation \( y = \pm \frac{b}{a} x \).
• Foci: (\( \pm c, 0 \)) contribute to defining the curve, with \( c \) derived using \( c^2 = a^2 + b^2 \).
• Directrices: Vertical lines at \( x = \pm \frac{a}{e} \) facing the vertices, offering a form of symmetrical balance.

These characteristics combine to describe the geometry and orientation of hyperbolas precisely, ensuring that each part of a hyperbola is accurately represented in graphs and equations.