Problem 23 - Calculus Early Transcendentals

Question

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=\cos t, y=1+\sin t ; 0 \leq t \leq 2 \pi$$

Step-by-Step Solution

Answer

Answer: The circle is described by the equation \(x^2 + (y - 1)^2 = 1\), with center \((0, 1)\) and radius 1. The orientation is counterclockwise, which is considered positive orientation.

Step 1: Write down the given parametric equations

The given parametric equations are $$x = \cos t$$ $$y = 1 + \sin t$$ with \(0 \leq t \leq 2 \pi\).

Step 2: Eliminate the parameter \(t\)

We will first square both equations and then add them together to eliminate the parameter \(t\). From the first equation, we have $$x^2 = \cos^2 t$$ From the second equation, we have $$y - 1 = \sin t$$ So, $$(y - 1)^2 = \sin^2 t$$ Now, we know that \(\sin^2 t + \cos^2 t = 1\), so we can add the two equations we found above: $$x^2 + (y - 1)^2 = 1$$

Step 3: Find the center and the radius

Now that we have the equation of the circle, we can identify the center and the radius. The equation is in the standard form $$(x - a)^2 + (y - b)^2 = r^2$$ Comparing this with our equation, we have $$x^2 + (y - 1)^2 = 1$$ We can see that the center of the circle \((a, b)\) is \((0, 1)\) and the radius \(r\) is equal to 1.

Step 4: Determine the positive orientation

Since the given parametric equation represents a full circle with \(0 \leq t \leq 2 \pi\), it traces the circle in a counterclockwise direction, which is considered the positive orientation.

Step 5: Final Answer

The circle can be described by the equation $$x^2 + (y - 1)^2 = 1$$ with center \((0, 1)\) and radius 1. It has a positive (counterclockwise) orientation.

Key Concepts

Circle Equation

A circle equation is a way to describe a circle on the coordinate plane. It connects the circle's geometric properties with an algebraic expression. The standard form of a circle's equation is \((x - a)^2 + (y - b)^2 = r^2\), where \(a, b\) is the center, and \(r\) is the radius.
In our example, we have the parametric equations \((x = \cos t)\) and \((y = 1 + \sin t)\). These describe the circle as \((x^2 + (y - 1)^2 = 1)\). This form directly reveals the circle's center and radius, fitting the standard circle equation perfectly.

Eliminating the Parameter

Eliminating the parameter in parametric equations involves finding a way to express \(x\) and \(y\) in terms of one another, without depending on a third variable \(t\). In our scenario, we want to remove \(t\) to find the equation relating just \(x\) and \(y\).
First, we square both equations: \(x^2 = \cos^2 t\) and \((y - 1)^2 = \sin^2 t\). By using the identity \(\sin^2 t + \cos^2 t = 1\), we add these results: \(x^2 + (y - 1)^2 = 1\). Now, we have a straightforward equation of a circle without the parameter \(t\).

Center and Radius of a Circle

Finding the center and radius from the circle's equation involves identifying key components in the standard form \( (x - a)^2 + (y - b)^2 = r^2 \).
In our derived equation, \(x^2 + (y - 1)^2 = 1\):

• The term \(x^2\) indicates that \(a = 0\).
• The term \((y - 1)^2\) means \(b = 1\).
• The equation \(= 1\) shows that the radius \(r = 1\).

Thus, the center is \( (0, 1) \) and the radius is \( 1 \). This information fully characterizes the geometry of the circle.

Positive Orientation

Orientation refers to the direction in which the circle is traced. In parametric equations, positive orientation usually means a counterclockwise direction.
For the given problem, with the range \(0 \leq t \leq 2\pi\), the equations trace the circle in a counterclockwise manner. This is because \(t\) progresses smoothly from \(0\) to \(2\pi\) without interruption, thus forming a complete loop following the natural increase of angle \(t\). Positive orientation is crucial for understanding the movement or direction along curves.