Problem 44 - Algebra and Trigonometry

Question

Convert the equation to polar form. $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

The polar form is \( r = 3 \).

Step 1: Understand the Problem

We need to convert the given Cartesian equation \(x^2 + y^2 = 9\) into polar form. Polar coordinates are represented by \(r\) and \(\theta\), where \(r\) is the radius and \(\theta\) is the angle.

Step 2: Recall Relationships between Cartesian and Polar Coordinates

The relationships between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\) are \(x = r \cos \theta\) and \(y = r \sin \theta\). Additionally, \(x^2 + y^2 = r^2\).

Step 3: Substitute into the Equation

Substitute \(x^2 + y^2 = r^2\) into the equation \(x^2 + y^2 = 9\). This gives us \(r^2 = 9\).

Step 4: Solve for r

To find \(r\), solve for \(r\) in the equation \(r^2 = 9\). This results in \(r = 3\) or \(r = -3\). However, since radius \(r\) is always non-negative in polar coordinates, we have \(r = 3\).

Step 5: Write the Final Polar Equation

The polar form of the equation is \(r = 3\). We do not need \(\theta\) as the angle is not restricted in the equation for a circle centered at the origin.

Key Concepts

Cartesian Coordinates

Cartesian coordinates form the basis of a two-dimensional coordinate system, known as the Cartesian plane. This system uses two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in this plane is described by an ordered pair \(x, y\). The first value, \(x\), represents the horizontal position of the point. The second value, \(y\), accounts for the vertical position.

There are several characteristics and uses of Cartesian coordinates:

• Understanding Position: Cartesian coordinates provide a straightforward way to understand the position of a point in space relative to the origin, which is \(0, 0\).


• Calculating Distance: Using the Pythagorean theorem, the distance between two points in the Cartesian plane can be easily calculated, aiding in various geometric problems.


• Plotting Graphs: Equations can be represented visually as lines, circles, or other shapes on the plane, with each form having a distinct visual shape.

In our original exercise, the equation \(x^2 + y^2 = 9\) represents a circle with a radius of 3 centered at the origin on the Cartesian plane.

Polar Form Conversion

Converting Cartesian equations to polar form allows us to express points considering both distance from a central point and an angle. Polar coordinates are given as \(r\) and \(\theta\), where \(r\) is the distance to the origin, and \(\theta\) is the angle relative to the positive x-axis.

If you want to convert from Cartesian to polar coordinates, remember:

• Correlation Formulas: Use the fundamental relationships: \(x = r \cos\theta\) and \(y = r \sin\theta\).


• Rewriting the Equation: Substitute these into the original Cartesian equation. For example, the Cartesian equation \(x^2 + y^2 = 9\) becomes \(r^2 = 9\) in polar form by replacing \(x^2 + y^2\) with \(r^2\).


• Simplifying: Solve for \(r\) to simplify the polar equation, such as deriving \(r = 3\) for a circle.

Polar conversions are particularly useful for identifying shapes with symmetry around a central point or axis, like circles and spirals.

Radius in Polar Coordinates

In polar coordinates, the radius \(r\) plays a central role as it indicates the distance of a point from the origin. Unlike Cartesian coordinates, where coordinates are based on a grid system, the radius in polar coordinates radiates out from a central point.

This makes the system:

• Central for Circles: In equations representing circles (such as \(r = 3\)), the radius directly gives the circle’s size with respect to a central reference point.


• Representing Shapes: Other geometric shapes can also be described uniquely by adjusting \(r\) and \(\theta\), offering insights into angles and curves.


• Non-negative Constraint: Radius in polar coordinates is always non-negative because it measures distance, which cannot be negative.

In our main exercise, \(x^2 + y^2 = 9\) converted directly into \(r^2 = 9\), revealing that the radius of the circle in this instance is a constant 3, implying that the circle is centered exactly at the origin with this radius.