Question

In Problems 67–70, the polar equation for each graph is either $r=a+b\cos \theta$or $r=a+b\sin \theta ,a>0$. Select the correct equation and find the values of $a$and $b$.

Short answer

The correct equation for the graph is $r=1+4\sin \theta$

And the value of a is $1$and bis$4$

Step-by-step solution

Step 1. Given

The graph:

The polar equations:

$$\begin{gathered} r=a+b\cos \theta \\ r=a+b\sin \theta \end{gathered}$$

Step 2. Test for r=a+b cos θ

Substitute the point $(1,0)$from the graph in the equation,

$$\begin{gathered} r=a+b\cos \theta \\ 1=a+b\cos \left(0\right) \\ 1=a+b \end{gathered}$$

Substitute the point $(5,\frac{\mathrm{\pi }}{2})$from the graph in the equation,

$$\begin{gathered} 5=a+b\cos \left(\frac{\mathrm{\pi }}{2}\right) \\ 5=a \end{gathered}$$

Step 3. Find b 

Substitute $a=5$in the equation,

$$\begin{gathered} 5+b=1 \\ b=-4 \end{gathered}$$

Since $b>0$, it is not a correct equation, so test the other equation.

Step 4. Test the equation r=a+b sin θ

Substitute $(1,0)$in the equation,

$$\begin{gathered} 1=a+b\sin \left(0\right) \\ a=1 \end{gathered}$$

Substitute $(5,\frac{\mathrm{\pi }}{2})$in the equation,

$$\begin{gathered} 5=1+b\sin \left(\frac{\mathrm{\pi }}{2}\right) \\ 5-1=b \\ b=4 \end{gathered}$$

Step 5. Find the equation

Substitute $a=1;b=4$in the suitable equation,

$r=1+4\sin \theta$

The polar equation of the given graph is $r=1+4\sin \theta$