Problem 32 - Precalculus Mathematics for Calculus

Question

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 10),\) vertices: \((0, \pm 8)\)

Step-by-Step Solution

Answer

\( \frac{y^2}{64} - \frac{x^2}{36} = 1 \)

Step 1: Understand the Hyperbola Structure

A hyperbola centered at the origin with a vertical transverse axis has the standard equation: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \], where \(a\) is the distance from the center to each vertex, and \(c\) is the distance from the center to each focus.

Step 2: Identify Given Values

Based on the problem, the vertices are given at \((0, \pm 8)\), so \(a = 8\), and the foci are \((0, \pm 10)\), so \(c = 10\).

Step 3: Use the Relationship between 'a', 'b', and 'c'

For a hyperbola, the relationship \(c^2 = a^2 + b^2\) holds. Substitute the given values: \(10^2 = 8^2 + b^2\).

Step 4: Solve for 'b'

Calculate \[100 = 64 + b^2\]. Subtract \(64\) from both sides to get \[b^2 = 36\]. So, \(b = 6\).

Step 5: Write the Equation of the Hyperbola

Substitute \(a^2\) and \(b^2\) into the standard equation of the hyperbola: \[ \frac{y^2}{8^2} - \frac{x^2}{6^2} = 1 \]. Simplify to get \[ \frac{y^2}{64} - \frac{x^2}{36} = 1 \].

Key Concepts

Conic Sections

Conic sections are fascinating geometric shapes formed by the intersection of a plane and a cone. There are four primary types: circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and equations. In this context, hyperbolas are our focus. A hyperbola consists of two disconnected curves called branches, and it has two focal points (foci) and a transverse axis connecting its vertices.

• Circles and Ellipses: These shapes are formed by cutting the cone parallel to its base or at an angle that does not pass through both nappes.
• Parabolas: Achieved when the cutting plane is parallel to one of the cone’s generatrices.
• Hyperbolas: Created when the plane intersects both nappes of the cone, resulting in an open curve.

Understanding conic sections is crucial in precalculus and analytical geometry, as it forms the foundation for more advanced mathematical concepts.

Precalculus Mathematics

Precalculus mathematics serves as the background knowledge needed to understand hyperbolas and other conic sections. It is a bridge between basic algebra, geometry, and calculus. In precalculus, you learn how to manipulate equations and understand their geometric implications, which is essential for deriving the equations of conic sections like hyperbolas.

• Equations and Graphs: Precalculus focuses on the relationships expressed in equations and how these equations can be converted into graphs.
• Trigonometry: Often, trigonometric identities help in understanding the angles involved in intersections of planes and cones.
• Functions and Modeling: These are used to describe and predict real-world phenomena using mathematical language.

By mastering these concepts, you are better prepared for advanced coursework, where you’ll explore the calculus and beyond.

Analytical Geometry

Analytical geometry, also known as coordinate geometry, combines algebra and geometry to explore geometric properties through algebraic equations. This blend allows for the analysis and characterization of geometric figures using a coordinate system. When dealing with hyperbolas, analytical geometry helps unfold their properties and derive their equations from given geometrical features like foci and vertices.

• Coordinate Systems: Provides a framework to place figures in a plane using coordinates, which is essential for defining hyperbolas.
• Distance and Midpoint Formulas: These are critical in calculating the properties of figures and understanding relationships between points.
• Deriving Equations: Analytical geometry uses known properties and relationships to derive equations representing hyperbolas and other figures. In this exercise, the equation \[c^2 = a^2 + b^2\] links the distances related to the foci and vertices.

These tools are invaluable for solving problems and are fundamental in engineering, physics, and many scientific disciplines.