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Chapter 10: Problem 45

In Exercises 41鈥�48, use the results of Exercises 37鈥�40 to find a set of parametric equations for the line or conic. Ellipse: vertices: \(( \pm 10,0) ;\) foci: \(( \pm 8,0)\)

Short Answer

Expert verified
The parametric equations are \(x = 10 \cdot cos(t)\) and \(y = 6 \cdot sin(t)\).

Step by step solution

01

Calculate the Distance from the Center to a Focus (f)

The foci of the ellipse are given as \(( \pm 8,0)\). The distances from the center of the ellipse to the foci are abs(-8) and abs(8), both of which equal to 8. So, \(f = 8\).
02

Find the Distance from the Center to a Vertex (a)

The vertices of the ellipse are given as \(( \pm 10,0)\). The distances from the center of the ellipse to the vertices are abs(-10) and abs(10), both of which equal to 10. So, \(a = 10\).
03

Calculate b

Knowing that \(a = 10\) and \(f = 8\), we calculate \(b\) using the formula \(b^2 = a^2 - f^2 = 10^2 - 8^2 = 36\). Thus, \(b = 6\).
04

Formulate the Parametric Equations

Our parametric representation of the ellipse will be \(x = a \cdot cos(t) = 10 \cdot cos(t)\) and \(y = b \cdot sin(t) = 6 \cdot sin(t)\). These are our parametric equations for the line or conic.

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Most popular questions from this chapter

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