# How do you graph r=8costheta?

Mar 17, 2017

Explained below

#### Explanation:

Write the given polar equation as ${r}^{2} = 8 r \cos \theta$

Nor convert to cartesean form ${r}^{2} = {x}^{2} + {y}^{2}$ and $r \cos \theta = x$, so that it is

${x}^{2} + {y}^{2} = 8 x$

${x}^{2} - 8 x + 16 - 16 + {y}^{2} = 0$

${\left(x - 4\right)}^{2} + {y}^{2} = 16$

This equation represents a circle with centre at (4,0) and radius 4. This can now be easily graphed

Mar 17, 2017

If you convert this equation to Cartesian coordinates, the resulting equation will be a circle.

#### Explanation:

Given: $r = 8 \cos \left(\theta\right)$

Multiply both sides by r:

${r}^{2} = 8 r \cos \left(\theta\right)$

Substitute the Cartesian conversion equations:

${x}^{2} + {y}^{2} = 8 x$

Add the $- 8 x + {h}^{2}$ to both sides:

${x}^{2} - 8 x + {h}^{2} + {y}^{2} = {h}^{2} \text{ }$

From the pattern ${\left(x - h\right)}^{2} = {x}^{2} - 2 h x + {h}^{2}$, we know that:

$- 2 h x = - 8 x$

$h = 4$

This makes the equation , become:

${\left(x - 4\right)}^{2} + {\left(y - 0\right)}^{2} = {4}^{2}$

This is a circle with a radius of 4 and a center at the Cartesian point $\left(4 , 0\right)$. Because the y coordinate is the polar point is the same, $\left(4 , 0\right)$

To graph the original equation, set your compass to a radius of 4 and put the center at the polar point $\left(4 , 0\right)$