Problem 19 - Thomas Calculus

Question

Sketch the circles in Exercises \(17-20 .\) Give polar coordinates for their centers and identify their radii. \(r=-2 \cos \theta\)

Step-by-Step Solution

Answer

The circle's center in polar coordinates is \((1, \pi)\) and the radius is \(1\).

Step 1: Understanding the Polar Equation

The given equation is \( r = -2 \cos \theta \). This equation suggests that the circle is expressed in polar coordinates. In polar form, equations of the type \( r = a \cos \theta \) or \( r = a \sin \theta \) are circles which are often centered on the polar axis.

Step 2: Converting to Cartesian Coordinates

To better understand the properties of the circle, we convert the polar equation to Cartesian coordinates. Using the relationships \( r = \sqrt{x^2 + y^2} \), \( x = r\cos\theta \), and \( y = r\sin\theta \), we can substitute: \( r = -2\cos\theta \Rightarrow r = -\frac{2x}{r} \). By rearranging and squaring both sides, we get: \( r^2 = -2x \). In terms of Cartesian coordinates, this becomes \( x^2 + y^2 = -2x \).

Step 3: Completing the Square

Let's complete the square for the equation \( x^2 + y^2 = -2x \). This involves shifting terms to find the center: \( x^2 + 2x + y^2 = 0 \). Rewrite as \( (x+1)^2 + y^2 = 1 \). Thus, the circle is centered at \((-1, 0)\) in Cartesian coordinates.

Step 4: Identify Center in Polar Coordinates

Given the center was found as \((-1, 0)\) in Cartesian, we convert this to polar coordinates. The radius \( r \) of the center is \(1\) and the angle \( \theta \) is \(\pi\) since it is on the negative x-axis. Thus, the polar coordinates are \((1, \pi)\).

Step 5: Determine the Radius

The equation in Cartesian form is \((x+1)^2 + y^2 = 1\). The radius of this circle is \(1\). In fact, this radius is equivalent to the absolute value of the constant in the polar equation \(-2\cos\theta\), thus confirming the radius is \(1\).

Step 6: Sketch the Circle

With center at \((1, \pi)\) in polar coordinates and radius \(1\), the circle is centered on the negative x-axis (since \(\theta = \pi\)) and extends one unit in all directions forming a circle. It slightly overlaps into the positive x direction by 1 unit.

Key Concepts

Polar Equations

In the context of polar coordinates, polar equations are used to describe curves by relating the radius \( r \) and the angle \( \theta \). The particular polar equation \( r = -2 \cos \theta \) defines a circle. Polar equations often take the form \( r = a \cos \theta \) or \( r = a \sin \theta \), which describes circles centered along certain axes. These equations are quite intuitive for circles, as they inherently incorporate angles and distances, which are crucial for defining circles. To visualize polar equations:

• They often use the polar formula for radius \( r \).
• The angle \( \theta \) determines the circle's orientation.
• The parameter \( a \) is closely related to the circle's properties like radius and center.

The equation \( r = -2 \cos \theta \) implies the circle is related to the x-axis due to the cosine function, with some properties like orientation or position being slightly adjusted because of the negative sign in front.

Cartesian Coordinates

To fully understand the geometry of a shape described by a polar equation, it is often useful to convert to Cartesian coordinates. This translates polar descriptions into the familiar \(x\) and \(y\) grid system.
In the given exercise, the polar equation \( r = -2 \cos \theta \) is converted into a Cartesian form. Leveraging basic polar-to-Cartesian conversions, we have:

• \( r = \sqrt{x^2 + y^2} \)
• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

When applied, the transformation showcases how expressions fit into familiar geometric contexts. From \( r^2 = -2x \) formed after conversion, it is found that the circle in Cartesian coordinates centers at \((-1, 0)\), adjusted through completing the square.

Circle Sketching

Sketching a circle defined by equations, especially between polar and Cartesian systems, involves determining the center and radius first. From the Cartesian form of the equation, \((x+1)^2 + y^2 = 1\), we extract essential details:

• Center is determined by solving for \(x\) and \(y\) positions, here \((-1, 0)\).
• Radius is identified as \(1\), as shown by the equation's form \((x+1)^2 + y^2 = 1\).

With this information:- Draw the circle centered at \(-1\) on the x-axis with a radius extending 1 unit in each direction.- The visualization shows it cutting across both sides of the x-axis, demonstrating its appearance when translated back to polar representations.
This process reflects how polar parameters translate into spatial dimensions.

Geometry

Geometry encompasses the study of shapes and their properties. In cases like this problem, geometry aids in deciphering different forms and understanding relationships between coordinates. Two systems—polar and Cartesian—provide diverse methods to describe shapes like circles.
Key geometrical insights from this task indicate:

• Center displacement, moving from polar descriptor \((1, \pi)\) to Cartesian \((-1, 0)\).
• Radius consistency, where the circle's extent is \(1\) unit from the center.
• Indicating circle's symmetry about this determined central point.

This reveals how varied mathematical approaches can signify similar geometric constructs. Each system, polar or Cartesian, brings unique perspectives to understanding a shape's nature.