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)

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Identify the center of the hyperbola by finding the midpoint of the vertices. Since the vertices are at (4,0) and (-4,0), the center is at \(\left( \frac{4 + (-4)}{2}, \frac{0 + 0}{2} \right) = (0,0)\).

Determine the distance from the center to each vertex, which is the value of \(a\). Here, \(a = 4\) because the vertices are 4 units away from the center along the x-axis.

Determine the distance from the center to each focus, which is the value of \(c\). Here, \(c = 6\) because the foci are 6 units away from the center along the x-axis.

Calculate \(b\) using the relationship for hyperbolas: \(c^2 = a^2 + b^2\). Rearranged, this is \(b = \sqrt{c^2 - a^2}\). Substitute the values of \(a\) and \(c\) to find \(b\).

Write the parametric equations for the hyperbola centered at the origin with a horizontal transverse axis: \(x = a \sec t\) \(y = b \tan t\) where \(t\) is the parameter.

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Parametric Equations of Conic Sections

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For conic sections like hyperbolas, these equations allow a clear representation of the curve's shape and position, facilitating analysis and graphing.

Properties of a Hyperbola

A hyperbola is defined by its vertices and foci, which determine its shape and orientation. The distance between vertices gives the transverse axis length, while the foci relate to the hyperbola's eccentricity, crucial for forming its standard equation and parametric form.

Relationship Between Vertices, Foci, and Parameters a, b, c

In a hyperbola, a represents the distance from the center to each vertex, c the distance to each focus, and b relates to the conjugate axis. These parameters satisfy the equation c² = a² + b², which is essential for deriving the hyperbola's parametric equations.

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)