Problem 23 - Algebra 2

Question

Write the equation of a hyperbola with the given foci and vertices. foci \(( \pm 5,0),\) vertices \(( \pm 3,0)\)

Step-by-Step Solution

Answer

The equation of the hyperbola with the given foci and vertices is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).

Step 1: Identify the center, a, and c values

The center of the hyperbola is at the origin (0,0) since the vertices and foci lie on the x-axis and are equidistant from the origin. The distance from the center to a vertex (a) is 3 units and the distance from the center to a focus (c) is 5 units.

Step 2: Calculate the b value

Knowing that the relationship between 'a', 'b' and 'c' is given by \(c^2 = a^2 + b^2\), we can plug in values of 'a' and 'c' to solve for 'b'. Hence, the equation becomes \(5^2 = 3^2 + b^2\). Calculating \(b^2\), we get \(b^2 = 5^2 - 3^2 = 16\), so, \(b = 4\).

Step 3: Write the equation of the hyperbola

Using the standard form for the equation of a hyperbola oriented horizontally, \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), and plugging in the values found (h=0, k=0, a=3, b=4), the equation of the hyperbola becomes \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).

Key Concepts

Equation of a Hyperbola

A hyperbola is a type of conic section that is characterized by its two branches, which are mirror images of each other. Its equation can take different forms, but in a standard Cartesian coordinate system, it is most commonly written as \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) for a horizontal orientation and \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \) for a vertical orientation.
This equation represents a hyperbola centered at the point \((h, k)\). The values \(a\) and \(b\) indicate the distances to the vertices and the co-vertices along the major and minor axes respectively.
In the example exercise, since the vertices \((\pm 3,0)\) and foci \((\pm 5,0)\) are on the x-axis, we use the form \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \), which denotes a horizontally oriented hyperbola.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. There are four primary types: circles, ellipses, parabolas, and hyperbolas.
Each conic section is defined by the angle and position of the intersecting plane. Hyperbolas occur when the plane cuts through both halves of the cone, resulting in two separate curves.
This conic's defining characteristic is that the difference of the distances from any point on the hyperbola to the two foci is constant. Hyperbolas have unique applications in fields such as astronomy and physics, where they help describe phenomena like the orbits of celestial bodies.

Hyperbola Properties

Hyperbolas have several properties that distinguish them from other conic sections. These properties include the axes, asymptotes, and orientation of the hyperbola.

• The **transverse axis** is the line that passes through the two foci.
• The **conjugate axis** is perpendicular to the transverse axis and passes through the center.
• The **asymptotes** are the lines that the hyperbola approaches but never touches.

The equation of the asymptotes for a hyperbola centered at the origin is given by \( y = \pm \frac{b}{a}x \). These asymptotes provide a frame within which the branches of the hyperbola reside.
Additionally, hyperbolas can be oriented horizontally or vertically, depending on the placement of the transverse axis.

Foci and Vertices

The foci and vertices are key components of a hyperbola's geometry which determine its shape and size. **Foci (singular: focus)** are two specific points inside each branch of the hyperbola.
The mathematical characteristic of a hyperbola is that the absolute difference of the distances from any point on the hyperbola to the two foci remains constant.
**Vertices** are the points where each branch of the hyperbola is closest to the center. These points determine the length of the transverse axis.

• For a hyperbola centered at \((h, k)\), the vertices are located at \((h\pm a, k)\)
• The foci are located at \((h\pm c, k)\), where \(c\) is found using the relation \(c^2 = a^2 + b^2\)

In our example, because the hyperbola is centered at the origin and aligned horizontally, the vertices and foci are given directly on the x-axis.