Problem 102 - A Graphical Approach to Precalculus with Limits

Question

Equation of a Hyperbola Use the definition of a hyperbola to find an equation of a hyperbola with center at the origin, foci \((-2,0)\) and \((2,0),\) and the absolute value of the difference of the distances from any point of the hyperbola to the two foci equal to 2 .

Step-by-Step Solution

Answer

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

Step 1: Understand the Hyperbola

A hyperbola is defined as the set of all points where the absolute difference of distances to two fixed points (the foci) is a constant. For the given hyperbola, this constant difference is 2.

Step 2: Identify the Center and Foci

The center of the hyperbola is at the origin (0, 0), and the foci are at points (-2, 0) and (2, 0). This indicates that the transverse axis is along the x-axis.

Step 3: Determine the Value of \(c\)

The distance from the center to each focus is called \(c\). Since the center is at (0,0) and the foci are at (-2,0) and (2,0), we find \(c = 2\).

Step 4: Apply the Hyperbola Equation Formula

The standard form of the equation of a hyperbola centered at the origin with a transverse axis along the x-axis is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Here, \(c^2 = a^2 + b^2\), and the given |distance from the foci difference| is 2, so \(2a = 2\).

Step 5: Solve for \(a^2\) and Compute \(b^2\)

From \(2a = 2\), we find \(a = 1\), and thus \(a^2 = 1\). Using \(c^2 = a^2 + b^2\), we substitute to find \(b^2\). Therefore, \(4 = 1 + b^2\) and \(b^2 = 3\).

Step 6: Write the Equation

Substitute \(a^2\) and \(b^2\) into the standard formula to obtain the equation: \(\frac{x^2}{1} - \frac{y^2}{3} = 1\).

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double napped cone. These curves are an essential part of geometry and are classified into four types:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

Each type of conic section has its own unique properties and equations. Hyperbolas, in particular, are defined as the set of points where the absolute difference of distances to two fixed points (known as foci) remains constant.
This property makes hyperbolas distinct from the other conic sections. Each conic section can provide profound insights into the geometric nature of various phenomena in mathematics, physics, and engineering.

Foci

Foci are specific points related to conic sections that crucially define their geometric nature. In a hyperbola, the foci hold a special importance. The hyperbola is defined such that the absolute difference in distances from any point on the hyperbola to the two foci is a constant, which is essential to understanding its shape.
For example, in the exercise above, the foci are positioned at (-2, 0) and (2, 0).
These locations directly influence the equation of the hyperbola, and understanding their role is key to solving related problems. The foci appear symmetrically around the center of the hyperbola, emphasizing its bilateral symmetry.

Equation of Hyperbola

The equation of a hyperbola is derived based on the geometry of the curve and takes different forms depending on its orientation and center. For a hyperbola centered at the origin (0, 0) with a transverse axis along the x-axis, the standard equation is:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]Where \(a\) and \(b\) are constants that represent distances related to the geometry of the hyperbola.
From the exercise, we learned that \(c\), the distance from the center to each focus, can be used alongside \(a\) and \(b\) through the relation \(c^2 = a^2 + b^2\). By following these steps, we see how altering the equation captures the essence of the hyperbola's unique geometry.

Transverse Axis

The transverse axis is an important component of a hyperbola. It is the line that passes through the center and the foci, defining the primary direction of the hyperbola’s opening.
In the given exercise, the transverse axis lies along the x-axis, as indicated by the foci (-2, 0) and (2, 0).
This axis determines that the hyperbola opens horizontally. By recognizing the transverse axis, we can accurately apply the correct formula for a hyperbola centered at the origin, leading to the equation's derivation. This foundational understanding helps in exploring deeper properties of hyperbolas in various mathematical contexts.