Problem 102 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Components of the Hyperbola

A hyperbola with a center at the origin and foci \((-2, 0)\) and \((2, 0)\) indicates that the transverse axis is horizontal. The distance between the foci \(2c\) is \(2 - (-2) = 4\), implying \(c = 2\). Given that the absolute value of the difference of the distances from any point on the hyperbola to the two foci is \(2\), this is the fixed value \(2a\), so \(a = 1\).

Step 2: Determine 'a' and 'b' Values

From the previous step, we have \(a = 1\). The relationship between \(a\), \(b\), and \(c\) in the standard form of the equation of a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) is given by \(c^2 = a^2 + b^2\). With \(c = 2\) and \(a = 1\), solve for \(b^2\):\[c^2 = a^2 + b^2 \Rightarrow 4 = 1 + b^2 \Rightarrow b^2 = 3\]

Step 3: Write the Equation of the Hyperbola

Substituting \(a^2 = 1\) and \(b^2 = 3\) into the standard form equation for a hyperbola centered at the origin with a horizontal transverse axis, we obtain:\[\frac{x^2}{1} - \frac{y^2}{3} = 1\]Thus, the equation of the hyperbola is \(x^2 - \frac{y^2}{3} = 1\).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle at which the plane cuts through the cone, you can form different types of curves. The main types of conic sections include:

• Ellipse: Formed when the plane intersects the cone at an angle less than that of the cone's side, but not parallel.
• Parabola: Formed when the plane is parallel to the edge of the cone.
• Hyperbola: Formed when the plane intersects both halves (nappes) of the cone.
• Circle: A special case of an ellipse when the cut is perpendicular to the cone's axis.

A hyperbola, specifically, is the curve formed when the intersecting plane makes a steep angle through both nappes, creating two opposite-facing open curves. Understanding these basic notions helps in visualizing the properties and equations associated with hyperbolas and other conics.

Hyperbola Properties

Hyperbolas have distinctive properties that set them apart from other conic sections. Here are a few important ones which are helpful in both understanding and solving hyperbola-related problems:

• Center: This is the midpoint between the foci.
• Foci: Two fixed points, located outside each arm of the hyperbola, which determine the curvature of the hyperbola.
• Vertices: These are the points where the hyperbola intersects its transverse axis.
• Transverse Axis: The line that passes through both foci and vertices. It is horizontally placed in this exercise.
• Asymptotes: Lines that the hyperbola approaches but never meets; they extend diagonally from the center.
• Eccentricity: A measure of how much the conic section deviates from being circular. Hyperbolas always have an eccentricity greater than 1.

For example, when given the foci and knowing that the distance difference from any point on the hyperbola to the foci is constant, you are beginning to uncover the hyperbola's fundamental shape.

Standard Form of Hyperbola

The standard form of a hyperbola's equation is crucial for understanding its geometric properties. For hyperbolas centered at the origin, the equation can take two primary forms based on its orientation:For a horizontally oriented hyperbola:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]And for a vertically oriented hyperbola:\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]The values of \(a\) and \(b\) affect the stretch of the hyperbola along the x and y axes respectively.

• \(a\): Represents the distance from the center to each vertex along the transverse axis.
• \(b\): Determines the asymptote slope, connecting with \(a\) and \(c\) in \(c^2 = a^2 + b^2\).
• \(c\): The distance from the center to each focus, where \(c^2 = a^2 + b^2\).

Knowing the standard form allows for direct translation into visual comprehension: from identifying the orientation of the hyperbola to calculating distances within its geometry.