In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.


Hyperbola: Vertices: (4,0) and (−4,0); Foci: (6,0) and (−6,0)

Verified step by step guidance

1. Identify the standard form of the hyperbola centered at the origin with horizontal transverse axis: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
2. Determine the value of \( a \) using the distance between the vertices: \( 2a = 8 \), so \( a = 4 \).
3. Determine the value of \( c \) using the distance between the foci: \( 2c = 12 \), so \( c = 6 \).
4. Use the relationship \( c^2 = a^2 + b^2 \) to find \( b^2 \): \( 6^2 = 4^2 + b^2 \).
5. Write the parametric equations for the hyperbola: \( x = a \sec(t) \) and \( y = b \tan(t) \), where \( t \) is the parameter.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a variable, typically time. For conic sections like hyperbolas, these equations allow us to describe the shape and position of the curve in a more flexible way than standard Cartesian equations. Understanding how to derive and manipulate these equations is essential for solving problems involving curves.

Hyperbola Properties

A hyperbola is defined as the set of all points where the difference of the distances to two fixed points (foci) is constant. Key properties include the location of the vertices, which are the points where the hyperbola intersects its transverse axis, and the foci, which are used to determine the shape and orientation of the hyperbola. Recognizing these properties is crucial for constructing the parametric equations.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone, resulting in shapes such as circles, ellipses, parabolas, and hyperbolas. Each type has distinct characteristics and equations. Understanding the general forms and properties of these conic sections is vital for identifying and working with their parametric representations.