#### Answer

The graphs of $r=a+b\sin \theta $, $r=a-b\sin \theta $, $r=a+b\cos \theta $, and $r=a-b\cos \theta $, $a>0$, $n>0$, are called limaçons, a French word for snail. The ratio $\frac{a}{b}$ determines the graph’s shape. If, $\frac{a}{b}=1$, the graph is shaped like a heart and called a cardioid. If $\frac{a}{b}<1$, the graph has an inner loop.

#### Work Step by Step

For the graph of the given function $r=a+b\sin \theta $, $r=a-b\sin \theta $, $r=a+b\cos \theta $, and $r=a-b\cos \theta $, $a>0$, $n>0$, the ratio of $\frac{a}{b}$ determines the shape of the graph. As the ratio of $\frac{a}{b}$ changes from zero to one, the graph changes it shape from an inner loop to a cardioid and as the ratio of $\frac{a}{b}$ increases beyond one, the shape of the graph is converted into a snail-like shape.