Question

Find the equation of the hyperbola defined by the given information. Sketch the hyperbola. Foci: (±3,0)\(;\) vertices: (±2,0)

Short answer

The hyperbola's equation is \( \frac{x^2}{4} - \frac{y^2}{5} = 1 \).

Step-by-step solution

Understand the Standard Form of a Horizontal Hyperbola

The standard form of a horizontal hyperbola centered at the origin is \( x^2/a^2 - y^2/b^2 = 1 \). The vertices are at \((\pm a, 0)\) and the foci are at \((\pm c, 0)\).

Identify Values of a and c

From the given vertices \((\pm 2, 0)\), we identify \( a = 2 \). From the given foci \((\pm 3, 0)\), we identify \( c = 3 \).

Determine b Using the Relationship c^2 = a^2 + b^2

Use the relationship \( c^2 = a^2 + b^2 \) to find \( b \). Substituting the known values, we have \( 3^2 = 2^2 + b^2 \). This simplifies to \( 9 = 4 + b^2 \), so \( b^2 = 5 \).

Plug Values into the Standard Form Equation

Substitute \( a^2 = 4 \) and \( b^2 = 5 \) into the hyperbola equation: \( \frac{x^2}{4} - \frac{y^2}{5} = 1 \).

Sketch the Hyperbola

To sketch the hyperbola, plot the vertices at \( (\pm 2, 0) \) and sketch the asymptotes passing through the origin. The asymptotes have slopes \( \pm \frac{b}{a} = \pm \frac{\sqrt{5}}{2} \). Draw the hyperbola opening horizontally from each vertex, extending towards the foci.

Key concepts

Foci

In a hyperbola, the foci are two distinct points that are crucial to its shape and properties. They are located along the transverse axis of the hyperbola. For this specific hyperbola, the foci are at

• coordinates \((\pm 3, 0)\)

which means they sit out farther than the vertices, determining the hyperbola's stretch. The distance from the center of the hyperbola (which in this case is at the origin (0,0)) to each focus point is represented by the variable \(c\). To find \(c\) in this problem, we use the given foci at

• \((\pm 3, 0)\),

which tells us that \(c = 3\). This distance \(c\) plays an essential role in taking us to the next step of understanding the hyperbola's equation.

Vertices

Vertices are another key feature of a hyperbola. They mark the points where the hyperbola intersects the transverse axis and are closer to the center than the foci. For the given hyperbola, the vertices are located at:

• \((\pm 2, 0)\).

These points help define the shape of the hyperbola, as they are essentially the "turning points" of the curve. The distance from the center (the origin) to each vertex is denoted as \(a\). In this scenario, we are told

• that \(a = 2\),

giving us a measure of how wide the hyperbola is. This value \(a\) is crucial to forming the standard equation of the hyperbola and calculating the related asymptotes.

Standard Form

The standard form of the equation for a horizontal hyperbola is \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]. This format tells us a lot about the hyperbola's size and orientation. **In a horizontal hyperbola, the hyperbola opens left and right, which fits this equation.** The values \(a\) and \(b\) represent distances related to the vertices and the asymptotes, respectively. In this problem:

• \(a^2 = 4\)
• \(b^2 = 5\).

These values are substituted into the standard form equation to illustrate this specific hyperbola. For a standard hyperbola equation centered at the origin, the vertices, foci, and asymptotes can all be determined once the standard form is known.

Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches. They play a significant role in defining the orientation and "direction" of the hyperbola. For a horizontal hyperbola centered at the origin with a standard form of \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\),the equations of the asymptotes are given by the slopes: \( \pm \frac{b}{a} \). In this exercise:

• \(a = 2\)
• \(b = \sqrt{5}\).

Thus, the slopes of the asymptotes become \( \pm \frac{\sqrt{5}}{2}\). These lines pass through the origin and give us a visual guideline to sketch the hyperbola accurately. As such, asymptotes are incredibly helpful tools when visually representing or plotting a hyperbola on a graph.