Question

Graphing Polar Equations

Sketch a graph of the polar equation $r=2+\sec \theta$ (conchoid)

Short answer

The graph of the given polar equation is-

Step-by-step solution

Step 1. Given information

A polar equation is given as $r=2+\sec \theta$

Step 2. Concept used

For sketching the graph, first, find different values r for different$\theta$. After that, convert them into x and y by using equations-

$\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Once we have the points draw a graph by plotting them.

Step 3. Calculation

The polar equation is given as$r=2+\sec \theta$,

Let$\theta =0$,

$\begin{array}{l}r=2+\frac{1}{\cos \theta }\\ r=2+1\\ r=3\end{array}$

Now when we plug the values in the equations $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

We get, $\begin{array}{l}x=3\\ y=0\end{array}$

Similarly,

Let$\theta =1$,

$\begin{array}{l}r=2+\frac{1}{\cos \theta }\\ r=2+1.85\\ r=3.85\end{array}$

Let$\theta =-1$,

$\begin{array}{l}r=2+\frac{1}{\cos \theta }\\ r=3.85\end{array}$

Plot the points on graph-