# How do you graph r=(23)/(7-5sin theta)?

Oct 21, 2016

See Socratic graph and explanation.

#### Explanation:

Use $\frac{l}{r} = 1 + e \cos \left(\theta - \alpha\right)$, with e < 1, represents an ellipse

with a focus at the pole (0, 0) and major axis along $\theta = \alpha$,

The semi major axis $a = \frac{l}{1 - {e}^{2}}$.

This can be reorganized to the form

$\frac{\frac{23}{7}}{r} = 1 + \frac{5}{7} \cos \left(\theta + \frac{\pi}{2}\right)$ revealing that the graph is the

ellipse with a focus S(0, 0). e = 5/7, $\alpha = - \frac{\pi}{2}$ , a =

161/24 = 6.71, nearly, and .

semi minor axis

$b = \sqrt{l a} = \sqrt{\left(\frac{23}{7}\right) \left(\frac{161}{24}\right)} = \frac{23}{\sqrt{24}}$ = 4.7, nearly.

A short Table for tracing the ellipse.

$\left(r , \theta\right)$:

$\left(0 , \frac{23}{7}\right) \left(\frac{46}{9} , \frac{\pi}{6}\right) \left(\frac{23}{2} , \frac{\pi}{2}\right) \left(\frac{46}{9} , \frac{5}{6} \pi\right) \left(\frac{23}{7} , \pi\right)$

$\left(\frac{46}{19} , \frac{7}{6} \pi\right) \left(\frac{23}{12} , \frac{3}{2} \pi\right) \left(\frac{46}{19} , \frac{11}{6} \pi\right) \left(\frac{23}{7} , 2 \pi\right)$

See a Socratic graph. Note that the major axis ( length 13.42 ) is

along y-axis.

graph{(x^2+y^2)^0.5-23/7 -5/7 y=0[-10 10 -3 12]}