Problem 42 - Precalculus Mathematics for Calculus

Question

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 1),\) length of transverse axis: 1

Step-by-Step Solution

Answer

The hyperbola equation is \(4y^2 - \frac{4}{3}x^2 = 1\).

Step 1: Identify the structure of the hyperbola

For a hyperbola with vertical transverse axis centered at the origin (0, 0), the standard form of the equation is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

Step 2: Determine the value of 'a' from the transverse axis

The length of the transverse axis is given as 1. The length of the transverse axis for a hyperbola is equal to \(2a\). Therefore, \(2a = 1\). Solving for \(a\), we get \(a = \frac{1}{2}\).

Step 3: Calculate 'c' from the foci

The foci are given as \((0, \pm 1)\), which means the distance from the center (0, 0) to each focus is \(c = 1\).

Step 4: Use the relationship between 'a', 'b', and 'c'

For hyperbolas, the relationship is \(c^2 = a^2 + b^2\). Substituting \(a = \frac{1}{2}\) and \(c = 1\), we have:\[1^2 = \left(\frac{1}{2}\right)^2 + b^2\]\[1 = \frac{1}{4} + b^2\]Solving for \(b^2\), we find \(b^2 = \frac{3}{4}\).

Step 5: Write the equation of the hyperbola

We now have all values needed for the equation of the hyperbola. Using \(a^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\) and \(b^2 = \frac{3}{4}\), substitute into the standard form:\[\frac{y^2}{\frac{1}{4}} - \frac{x^2}{\frac{3}{4}} = 1\]This simplifies to:\[4y^2 - \frac{4}{3}x^2 = 1\]

Key Concepts

Transverse Axis

The transverse axis is a fundamental concept when dealing with hyperbolas. It refers to the line segment that passes through the center of the hyperbola and both its vertices. For a hyperbola centered at the origin (0, 0), like in the given problem, the transverse axis orientation determines whether the hyperbola opens horizontally or vertically.

- **Orientation:** A vertical transverse axis implies that the hyperbola opens upward and downward. Conversely, a horizontal transverse axis means the hyperbola opens to the left and right.- **Length:** The length of the transverse axis is denoted by two times the distance from the center to one of the vertices, expressed mathematically as \(2a\), where \(a\) is the distance from the center to each vertex along the transverse axis.

In the provided example, the transverse axis is vertical with a total length of 1, leading to the determination that \(a = \frac{1}{2}\). This length helps derive the hyperbola's equation by providing vital details about its shape.

Foci

The concept of foci is crucial in understanding the structure of any conic section, particularly hyperbolas. A hyperbola has two distinct foci which are located symmetrically outside its vertices. These points play a significant role in defining the shape and properties of the hyperbola.

- **Definition:** Each focus of a hyperbola is positioned at a distance \(c\) from the center, and it lies along the line that contains the transverse axis.- **Importance:** The difference in distances from any point on the hyperbola to the two foci is a constant value. This unique property distinguishes hyperbolas from other conic sections.

In the exercise's example, the foci are given as \(0, \pm 1\), indicating that \(c = 1\). The distance \(c\) aids in finding the other necessary parameters, such as \(a\) and \(b\), for constructing the equation of the hyperbola. Furthermore, understanding \(c\) in context with to foci is key to visualizing the hyperbola correctly.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double cone. They include several distinct figures such as circles, ellipses, parabolas, and hyperbolas. Among these, hyperbolas feature two curving, separate branches. The way these shapes and sections are classified based on the angle and position of the intersecting plane relative to the cone: - **Circle**: Formed when the plane cuts perpendicular to the axis of the cone. - **Ellipse**: Occurs when the plane slices at an angle, but does not intersect the base. - **Parabola**: Emerges when the plane is parallel to the slant of the cone. - **Hyperbola**: Appears when the plane cuts through both halves of the cone, resulting in two mirror-image curves. Hyperbolas can be easily recognized by their distinctive open, double-arc configurations and their unique properties governed by foci, directrices, and asymptotes. In the study of hyperbolas, comprehending these essential conic section concepts helps to effectively solve equations and predict curve behavior, paramount for students learning about these geometric structures.