Problem 19 - Essential Calculus

Question

Convert the rectangular equation to polar form and sketch its graph. $$ x^{2}+y^{2}=a^{2} $$

Step-by-Step Solution

Answer

The polar form of the given equation is \(r=|a|\). The graph is a circle with radius \(a\) if \(a>0\) and circle with radius \(-a\) from the opposite direction if \(a<0\) centered at the origin.

Step 1: Convert Rectangular Equation to Polar Form

Use the conversion relations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Substituting these into the equation results in \((r\cos(\theta))^{2}+(r\sin(\theta))^{2}=a^{2}\), which simplifies to \(r^{2}=a^{2}\). Taking square root of both sides gives \(r=|a|\), hence, the polar form is \(r=a\) for \(a>0\) and \(r=-a\) for \(a<0\).

Step 2: Graphing the Polar Equation

The graph of \(r=a\) is a circle centered at the origin with a radius of \(a\), if \(a>0\), and likewise \(r=-a\) is also a circle centered at the origin, but is conventionally represented in polar coordinates by the negative radius under the graph \(r=a\). Therefore, if \(a<0\), then move in the opposite direction to the initial direction that theta indicates.