Problem 32 - Calculus Early Transcendentals

Question

Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y .\) Answers are not unique. A circle centered at \((2,-4)\) with radius \(\frac{3}{2},\) generated counterclockwise with initial point \(\left(\frac{7}{2},-4\right)\)

Step-by-Step Solution

Answer

Answer: The parametric equations for the given circle are \(x = 2 + \frac{3}{2}\cos t\) and \(y = -4 + \frac{3}{2}\sin t\) with the parameter values in the interval \([0, 2\pi)\). The circle can be described in terms of x and y as \((x-2)^2+(y+4)^2=\frac{9}{4}\).

Step 1: Find the general equation of the circle

The general equation of a circle is \((x-a)^2+(y-b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius. So, in this case, the general equation is: \((x-2)^2+(y+4)^2=\left(\frac{3}{2}\right)^2\)

Step 2: Find the parametric equations of the circle

To find the parametric equations, we express x and y in terms of a parameter t (angle) using trigonometry. The initial point is given as \(\left(\frac{7}{2}, -4\right)\). Let \(t\) be the angle generated counterclockwise from the initial point, so the parametric equations are: \(x = 2 + \frac{3}{2}\cos t\) and \(y = -4 + \frac{3}{2}\sin t\)

Step 3: Interval for parameter values

As the parameter t represents the angle, we can define its interval as \(0 \le t < 2\pi\), where \(0\) is inclusive and \(2\pi\) is exclusive. This interval represents one full rotation counterclockwise around the circle.

Step 4: Describe the circle in terms of x and y

We already have the general equation from Step 1, so we can rewrite it as: \((x-2)^2+(y+4)^2=\frac{9}{4}\) To summarize, the parametric equations for the given circle are: \(x = 2 + \frac{3}{2}\cos t\) and \(y = -4 + \frac{3}{2}\sin t\) with the parameter values in the interval \([0, 2\pi)\). The circle can be described in terms of x and y as: \((x-2)^2+(y+4)^2=\frac{9}{4}\).

Key Concepts

Trigonometric Representation of a Circle

When we deal with circles in a coordinate system, their geometric nature can be beautifully captured using trigonometry. Specifically, the trigonometric representation of a circle allows us to express the position of any point on a circle moving counterclockwise as a function of an angle.

In this representation, assuming a circle with its center at \<(a,b)\> and radius \(r\), any point \(P\) on the circle can be described using the angle \(t\) as follows:

• \(x = a + r \cdot \cos(t)\)
• \(y = b + r \cdot \sin(t)\)

The angle \(t\) is measured from the positive x-axis direction towards the point where the circle starts being traced out, often referred to as the initial point. This angle increases counterclockwise, which is considered the positive direction in mathematics and the trigonometric functions \(\cos(t)\) and \(\sin(t)\) provide the coordinates of the point relative to the circle's center.

In our exercise example, the parametric equations translated the initial point \(\left(\frac{7}{2},-4\right)\) into a set of equations by incorporating the angle \(t\), starting at the initial point and moving counterclockwise.

• \(x = 2 + \frac{3}{2}\cos(t)\)
• \(y = -4 + \frac{3}{2}\sin(t)\)

This approach is particularly powerful for modeling circular motion and understanding the position of points on a circle in many applied sciences and engineering disciplines.

General Equation of a Circle

Every high school geometry student encounters the standard form of the equation of a circle: \((x - a)^2 + (y - b)^2 = r^2\), which provides a straightforward method to describe a circle in a Cartesian coordinate system. Here, \((a, b)\) represents the center of the circle, and \(r\) is the radius, the constant distance from the center to any point on the circle's edge.

We can use this general equation to plot the circle's position on the graph or to find key properties like its area or circumference. For example, in the solved exercise, the circle's center and radius were identified, and the general equation was used to express the shape of the circle:

• \((x-2)^2+(y+4)^2=\left(\frac{3}{2}\right)^2\)

Understanding and being able to manipulate this equation is essential for students as it lays the foundation for more complex topics in geometry and calculus, such as calculating tangents, normals, and areas bounded by circular curves.

Polar Coordinates

The polar coordinate system offers an alternative to the Cartesian coordinate system for describing positions of points in a plane. Instead of using \(x\) and \(y\) coordinates, polar coordinates use a radius \(r\) and an angle \(\theta\), which are analogous to the distance from a point to the origin and the counterclockwise angle measured from the positive x-axis to the point.

Conversion from Cartesian coordinates to polar coordinates and vice versa is commonly achieved using the relationships:

• \(x = r \cdot \cos(\theta)\)
• \(y = r \cdot \sin(\theta)\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan\left(\frac{y}{x}\right)\), for \(x > 0\)

When dealing with circles, using polar coordinates can simplify equations and calculations, especially in fields like physics and engineering where rotational dynamics are analyzed. For instance, the parametric equations from our exercise can also be interpreted in polar coordinates, connecting the ideas of radius, angle, and movement along a circular path.