Question

consider the limaçon with equation $r = 3 + 4cos(\theta)$. how does the quotient of a and b relate to the existence of an inner loop? because $\frac{a}{b}>1$, the curve is a limaçon with an inner loop. because $\frac{b}{a}>1$, the curve is a limaçon with an inner loop. because $\frac{a}{b}>1$, the curve is a limaçon without an inner loop. because $\frac{b}{a}>1$, the curve is a limaçon without an inner loop

Explanation:

Step1: Identify the form of the limacon

The general polar - equation of a limacon is $r=a + b\cos(\theta)$. Here $a = 3$ and $b = 4$.

Step2: Recall the condition for an inner - loop

The condition for a limacon $r=a + b\cos(\theta)$ to have an inner - loop is $\frac{b}{a}>1$.

Step3: Calculate the ratio

Calculate $\frac{b}{a}=\frac{4}{3}>1$. So the limacon $r = 3+4\cos(\theta)$ has an inner loop.

Answer:

For the limacon of the form $r = a + b\cos(\theta)$ (in our case $r=3 + 4\cos(\theta)$, so $a = 3$ and $b=4$), when $\frac{b}{a}>1$, the curve is a limacon with an inner - loop. So the answer is: Because $\frac{b}{a}>1$, the curve is a limacon with an inner loop.