Problem 64 - A Graphical Approach to College Algebra

Question

Find an equation for each hyperbola. \(y\) -intercepts \((0, \pm 5) ;\) foci \((0, \pm 3 \sqrt{3})\)

Step-by-Step Solution

Answer

Equation: \(\frac{y^2}{25} - \frac{x^2}{2} = 1\).

Step 1: Understanding Hyperbola Orientation

Given that the hyperbola has y-intercepts and focuses aligned vertically, this suggests a hyperbola centered at the origin with the y-axis as the major axis. The equation takes the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Step 2: Identifying 'a' from y-intercepts

The y-intercepts are \((0, \pm 5)\). This implies that \(a = 5\), since the vertices are located at \((0, \pm a)\). Hence, \(a^2 = 25\).

Step 3: Identifying 'c' from Foci

The foci are \((0, \pm 3 \sqrt{3})\). This implies that \(c = 3 \sqrt{3}\). Therefore, \(c^2 = 27\).

Step 4: Using Relationship Between a, b, and c

For hyperbolas, the relationship is \(c^2 = a^2 + b^2\). Substituting the known values: \(27 = 25 + b^2\).

Step 5: Solving for b^2

Rearrange the equation from the previous step: \(b^2 = 27 - 25 = 2\).

Step 6: Writing the Standard Form Equation

Now that all values are determined, insert them into the standard form: \(\frac{y^2}{25} - \frac{x^2}{2} = 1\).

Key Concepts

Conic Sections

Conic sections are a fascinating topic in mathematics, and they arise from slicing through a cone at different angles. These include various shapes like circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and equations that describe their structure.
Hyperbolas are one of these conic sections and are formed when a plane cuts through both nappes (the upper and lower regions) of the cone. This cutting does not go parallel to the axis, resulting in two separate curves that open away from each other. This shape can be likened to two mirrored parabolas.

• Conic sections can be circular, elliptical, parabolic, or hyperbolic.
• They appear in different scientific fields, showcasing their applicational importance.
• Understanding these shapes provides insight into geometry and algebra, giving a bridge between abstract math and real-world phenomena.

Exploring hyperbolas exposes students to an interesting blend of geometric intuition and algebraic representation. This makes the study of conic sections both fulfilling and central to further mathematical learning.

Standard Form of Hyperbola

The standard form of a hyperbola's equation helps us describe its size, position, and orientation on the coordinate plane. For hyperbolas, these equations can vary depending on whether they open horizontally or vertically.
In this case, the equation is structured as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). This form is used when the hyperbola opens up and down along the y-axis with its vertex at the origin. Here are some key details:

• The variable \(y\) is over \(a^2\), showing that the hyperbola opens along the y-axis.
• \(a\) represents the distance from the center to each vertex along the y-axis.
• \(b^2\) is derived from the relationship \(c^2 = a^2 + b^2\), where \(c\) is the distance from the center to each focus.

This standard form is fundamental as it lays down the algebraic representation of hyperbolas. Recognizing hyperbolas through this format is essential for graphing and further geometric comprehension.

Foci of Hyperbola

In hyperbolas, identifying the foci is crucial as they provide a deeper understanding of the structure's geometry. The foci are two fixed points located along the axis of the hyperbola, and the distance between any point on the hyperbola to these foci is a consistent mathematical relationship.
The given problem specifies the foci at \((0, \pm 3 \sqrt{3})\), indicating that this hyperbola is vertically oriented. Here's what you need to know:

• The foci determine the "spread" of the hyperbola, affecting how wide or narrow the curves are.
• This is encapsulated by \(c\), where \(c = \sqrt{a^2 + b^2}\).
• Understanding the foci helps in plotting the hyperbola accurately on a coordinate plane, providing a reference for locating the curve in space.

The concept of foci is foundational to comprehending complex shapes like hyperbolas because they encapsulate the definition of these forms. It's crucial to leverage these focal points to gain an intuitive grasp of this wonderful geometric shape.