Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x^{2}+y^{2}=2 $$

Step-by-Step Solution

Answer

The polar equation is \(r = \sqrt{2}\).

Step 1: Understand the Equation Type

The given Cartesian equation is a circle equation: \(x^2 + y^2 = 2\). This represents a circle centered at the origin with a radius of \(\sqrt{2}\).

Step 2: Recall Polar to Cartesian Transformations

In polar coordinates, the transformations are: \(x = r\cos\theta\) and \(y = r\sin\theta\). Moreover, \(r^2 = x^2 + y^2\).

Step 3: Substitute Cartesian Terms with Polar Equivalents

Substitute \(x^2 + y^2\) with \(r^2\) in the polar equation to transform the given Cartesian equation. Thus, the equation becomes \(r^2 = 2\).

Step 4: Solve for the Polar Equation

Since \(r^2 = 2\), taking the square root of both sides gives \(r = \sqrt{2}\). This is the polar equation for the given Cartesian circle.

Key Concepts

Cartesian to Polar Transformation

Transforming equations from Cartesian coordinates to polar coordinates involves a few straightforward substitutions. To establish a connection between Cartesian and polar coordinates:

• Replace the Cartesian coordinates \( x \) and \( y \) with the polar coordinates' equivalent forms: \( x = r\cos\theta \) and \( y = r\sin\theta \).
• Use \( r^2 = x^2 + y^2 \) to link the coordinates.

These transformations are used to express relationships in scenarios that involve radial symmetry or angles, such as circular shapes or rotations.
In the case of our example, the Cartesian circle equation \( x^2 + y^2 = 2 \) can seamlessly transform to the polar form \( r^2 = 2 \). This transformation is possible because polar coordinates naturally describe circles with \( r \) referring to the radius, emphasizing the symmetry and simplicity polar coordinates can bring.

Circle Equation

Circle equations are a prime example of how to model symmetrical shapes. In Cartesian coordinates, circles are usually expressed as \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle. This form indicates that every point on the circle is equidistant from the center, determined by the radius.
For the given example \( x^2 + y^2 = 2 \), the equation implies a circle centered at the origin with a radius of \( \sqrt{2} \).

• This equation style is not only helpful for plotting but also offers insights into the circle's characteristics, like its radius and center.
• Being familiar with these properties can aid in converting the circle from one coordinate system to another.

In polar coordinates, it simplifies further due to the inherent manner polar coordinates reference radial and angular measurements.

Polar Coordinates

Polar coordinates can describe a point in the plane using a distance and an angle, rather than two perpendicular distances.

• The distance from the origin is called \( r \) (the radial coordinate).
• The angle \( \theta \), measured from the positive x-axis, is the angular coordinate.

This system is pivotal when dealing with problems that involve cycles, rotations, and symmetry around a point, such as circles or spirals.
In the exercise at hand, converting the circle equation to polar form resulted in the equation \( r = \sqrt{2} \). Here, \( r \) is constant, indicating every point on this circle lies at a steady radius from the origin. This characteristic displays the elegance and utility of polar coordinates in presenting circular geometries with simplicity, especially in scenarios where the origin serves as a natural focal point.