Problem 66 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(x^2+y^2=16\)

Step-by-Step Solution

Answer

The polar form of the given rectangular equation \(x^2+y^2=16\) is \(r = 4\).

Step 1: Identify the given rectangular equation

The given rectangular equation is \(x^2+y^2=16\).

Step 2: Write down the relationship between rectangular and polar coordinates

In polar coordinates, the relationships between the two coordinate systems are given by \(r^2 = x^2 + y^2\) and \(\theta = \tan^{-1} (y/x)\). However, for this conversion, only the first relationship is needed because the equation is symmetrical.

Step 3: Convert the rectangular equation into polar form

Substitute \(x^2+y^2\) with \(r^2\) in the given equation. This gives the polar form of the equation as \(r^2=16\).

Step 4: Simplify the polar form of the equation

Taking the square root of both sides, the polar form of the equation simplifies to \(r=4\) when \(r>0\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, is a system that allows us to locate points in a plane using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is identified as an ordered pair \(x, y\). The first value \(x\) is the distance from the y-axis, while the second value \(y\) is the distance from the x-axis. This method provides a clear and structured way to describe locations and shapes in 2-dimensional space. For example, the equation \(x^2+y^2=16\) represents a circle centered at the origin with a radius of 4.

Coordinate Conversion

Converting coordinates between different systems involves using mathematical relationships to transform points from one system to another. In the case of rectangular to polar conversion, we use the equations: \[\begin{align*}x &= r \cos \theta,\y &= r \sin \theta,\r^2 &= x^2 + y^2.\\end{align*}\]
This transformation is beneficial for simplifying the representation of curves and shapes, especially those symmetric about a point. The exercise is about converting the rectangular equation \(x^2+y^2=16\) to polar form. By substituting \(x^2+y^2\) with \(r^2\), we get the simpler equation \(r^2 = 16\), which further simplifies to \(r = 4\) when considering positive r-values.

Symmetrical Equations

Symmetrical equations are those that maintain their form despite transformations, such as translations or rotations. Symmetry can make equations easier to analyze and convert into different forms. For circular symmetry like in the exercise \(x^2+y^2=16\), converting to polar form highlights the symmetry about the origin. The equation \(r = 4\) in polar coordinates shows a set of points that are consistently at a distance of 4 from the origin, illustrating this circular symmetry clearly. Recognizing symmetrical equations helps streamline conversion processes and enhances understanding of geometric properties.