Problem 24 - Precalculus: Functions and Graphs

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 3)\) vertices \(V(0, \pm 2)\)

Step-by-Step Solution

Answer

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

Step 1: Identify Hyperbola Orientation

Since the foci and vertices are on the y-axis, this is a vertical hyperbola. The general equation for a vertical hyperbola centered at the origin is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

Step 2: Determine \(a\) from the Vertices

The distance from the center to each vertex is \(a\). Here, one vertex is \(V(0, \pm 2)\). Thus, \(a = 2\).

Step 3: Determine \(c\) from the Foci

The distance from the center to each focus is \(c\). Here, one focus is \(F(0, \pm 3)\). Thus, \(c = 3\).

Step 4: Calculate \(b\) using the Formula \(c^2 = a^2 + b^2\)

Using the values of \(a\) and \(c\), substitute into the equation \(c^2 = a^2 + b^2\) to find \(b^2\). Thus, \(3^2 = 2^2 + b^2\) gives \(b^2 = 9 - 4 = 5\).

Step 5: Formulate the Equation of the Hyperbola

Substitute \(a = 2\) and \(b = \sqrt{5}\) into the equation for a vertical hyperbola centered at the origin: \( \frac{y^2}{2^2} - \frac{x^2}{5} = 1 \), which simplifies to \( \frac{y^2}{4} - \frac{x^2}{5} = 1 \).

Key Concepts

Equations of Hyperbolas

In coordinate geometry, hyperbolas are fascinating conic sections with two separate branches. A hyperbola is defined by the differences between distances to two fixed points called foci. The general equation of a hyperbola can take one of two forms, depending on its orientation. For hyperbolas centered at the origin, these are:

• Horizontal Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• Vertical Hyperbola: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

These equations allow us to understand the shape and nature of the hyperbola based on the parameters \(a\) and \(b\). The constants \(a\) and \(b\) represent the distances from the center to the vertices and are crucial for sketching the graph of the hyperbola. The use of these equations aids in determining other features like the position of asymptotes and the exact path of the branches.

Foci and Vertices

Every hyperbola has two important sets of points: foci and vertices. The vertices lie along the transverse axis, while the foci exist further along the same axis, beyond the vertices. They each have distinct roles:

• Vertices: The points where the hyperbola intersects its principal axis. The distance from the center to a vertex is denoted by \(a\).
• Foci: These points are not on the hyperbola itself, but are key in defining it. The distance from the center to a focus is represented by \(c\).

The relationship between these elements is mathematically formulated as \(c^2 = a^2 + b^2\), a critical equation which helps in calculating all necessary dimensions for graphing a hyperbola accurately. Knowing the positions of foci and vertices is essential for crafting the precise graph of the hyperbola on a coordinate plane.

Distance in Coordinate Geometry

Understanding distances in coordinate geometry is vital for working with hyperbolas. The distance formula allows for the calculation between any two points, crucial for determining distances like those between a center and a vertex or focus.The distance formula is given by:\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]For hyperbolas, using this formula helps find necessary distances for vertices and foci, should they not be directly given. However, in our equation work, we often use simplified relationships like the vertex distance (\(a\)) and the focus distance (\(c\)) to simplify construction. These measurements are crucial for both graphing and understanding the spatial properties and symmetries of hyperbolas.

Vertical Hyperbola

A vertical hyperbola is one where the major axis (the transverse axis) is aligned vertically. This makes the y-component dominant in its orientation and appearance on a graph. The equation for a vertical hyperbola is:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]This denotes that the hyperbola stretches wider up and down rather than side-to-side. In practical terms, this means its vertices, as well as its foci, lie on the y-axis. This results in a vertically elongated shape.For our exercise where foci are at \(F(0, \pm 3)\) and vertices at \(V(0, \pm 2)\), we can clearly see the hyperbola's symmetry around the vertical axis. The acute understanding of the vertical versus horizontal orientation helps in discerning the correct mathematical approach needed for solving and graphically representing a hyperbola.