Problem 22 - Thomas Calculus

Question

Find polar equations for the circles in Exercises \(21-28 .\) Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. $$ (x+2)^{2}+y^{2}=4 $$

Step-by-Step Solution

Answer

Polar equation: \(r^2 + 4r\cos(\theta) = 0\); Cartesian: \((x + 2)^2 + y^2 = 4\).

Step 1: Identify Circle's Center and Radius

The given Cartesian equation of a circle is \((x + 2)^{2} + y^{2} = 4\). Here, the circle is centered at \((-2, 0)\) and has a radius of 2, since 4 is the square of the radius, \(r^{2} = 4\), thus \(r = 2\).

Step 2: Establish Polar Coordinates

Recall that in polar coordinates, \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). The center \((-2, 0)\) corresponds to polar coordinates \((r_c, \theta_c)\) where \(r_c = 2\) and \(\theta_c = \pi\) because the center is on the negative x-axis.

Step 3: Derive Polar Equation for the Circle

In polar form, a circle with center at \((r_0, \theta_0)\) and radius \(a\) is given by:\[ r^2 - 2r r_0\cos(\theta - \theta_0) = a^2 \]Substitute \(r_0 = 2\), \(\theta_0 = \pi\), and \(a = 2\) into this equation:\[ r^2 + 4r\cos(\theta) + 4 = 4 \]Simplifying, we have:\[ r^2 + 4r\cos(\theta) = 0 \]

Step 4: Verify and Sketch

Verify that the derived polar equation \(r^2 + 4r\cos(\theta) = 0\) matches the properties of the original Cartesian circle. The equation in polar form is consistent with the center at \((-2, 0)\) and radius 2.For the sketch: In Cartesian coordinates, the circle is centered at \((-2, 0)\) with radius 2; in polar form, the circle can be sketched using \(r = -2\cos(\theta)\) fulfilling the condition \(r^2 + 4r\cos(\theta) = 0\).

Key Concepts

Cartesians Equations

In mathematics, the Cartesian equation is a valuable form to represent geometric figures like circles, lines, and more. It uses x and y coordinates to define a shape on the coordinate plane. For example, the Cartesian equation of a circle

• Presents the formula \( (x-h)^2 + (y-k)^2 = r^2 \) where \( (h,k) \) is the center and \( r \) is the radius.
• In our specific exercise, \( (x+2)^2+y^2=4 \) reveals a circle centered at \((-2,0)\) with a radius of 2.

Understanding the conversion between Cartesian and polar equations allows one to better visualize the circle's placement and orientation on the graph. Each point on this circle satisfies the given equation, forming a fixed distance from the center.

Circle in Coordinate Plane

A circle in the coordinate plane is a set of all points equidistant from a center point. The equation \((x+2)^2 + y^2 = 4\) represents a circle that:

• Has a center at \((-2,0)\).
• Features a fixed radius of 2 units.

To graph this circle, start by plotting the center, then draw all points 2 units away from this center. Every point on this circle equates to a certain solution of the equation, showcasing symmetry about its center. Graphing tools can help visualize this well-defined shape, enabling ease in sketching it manually.

Polar Coordinates

Polar coordinates offer an alternative way of representing points on a plane using a distance and an angle rather than x and y positions. This system is particularly useful when working with shapes like circles that are naturally oriented around a center point. For a circle, the conversion process includes:

• Replacing \( x \) and \( y \) in the Cartesian equation with \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \).
• For the given circle, this leads to a polar equation, \(r^2 + 4r\cos(\theta) = 0\).

This form is beneficial when plotting and analyzing circular paths in contexts such as physics and engineering, where direction and distance are more intrinsic.

Mathematics Education

Mathematics education emphasizes understanding different ways of representing data, such as Cartesian and polar forms, to enrich comprehension. This practice assists students in:

• Acquiring a versatile thinking approach.
• Enhancing their problem-solving skills for more complex tasks.
• Encouraging interactive learning by applying multiple methods for a single concept.

Grasping connections between forms, as seen with the circle's equations in both Cartesian and polar formats, signifies a deeper understanding of coordinate systems and geometric properties. Actively practicing these conversions aids in cementing the foundational skills necessary for advanced mathematical studies.