# How do you convert r(2 - cosx) = 2 to rectangular form?

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Jan 29, 2017

The explanation is written with the assumption that you meant $\theta$ for the argument of the cosine function.

#### Explanation:

Use the distributive property :

$2 r - r \cos \left(\theta\right) = 2$

$2 r = r \cos \left(\theta\right) + 2$

Substitute x for $r \cos \left(\theta\right)$ and $\sqrt{{x}^{2} + {y}^{2}}$ for r:

$2 \sqrt{{x}^{2} + {y}^{2}} = x + 2$

Square both sides:

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

$4 {x}^{2} + 4 {y}^{2} = {x}^{2} + 4 x + 4$

$3 {x}^{2} - 4 x + 4 {y}^{2} = 4$

Add $3 {h}^{2}$ to both sides:

$3 {x}^{2} - 4 x + 3 {h}^{2} + 4 {y}^{2} = 3 {h}^{2} + 4$

Remove a factor of the 3 from the first 3 terms:

$3 \left({x}^{2} - \frac{4}{3} x + {h}^{2}\right) + 4 {y}^{2} = 3 {h}^{2} + 4$

Use the middle in the right side of the pattern ${\left(x - h\right)}^{2} = {x}^{2} - 2 h x + {h}^{2}$ and middle term of the equation to find the value of h:

$- 2 h x = - \frac{4}{3} x$

$h = \frac{2}{3}$

Substitute the left side of the pattern into the equation:

$3 {\left(x - h\right)}^{2} + 4 {y}^{2} = 3 {h}^{2} + 4$

Substitute $\frac{2}{3}$ for h and insert a -0 into the y term:

$3 {\left(x - \frac{2}{3}\right)}^{2} + 4 {\left(y - 0\right)}^{2} = 3 {\left(\frac{2}{3}\right)}^{2} + 4$

$3 {\left(x - \frac{2}{3}\right)}^{2} + 4 {\left(y - 0\right)}^{2} = \frac{16}{3}$

Divide both sides by $\frac{16}{3}$

$3 {\left(x - \frac{2}{3}\right)}^{2} / \left(\frac{16}{3}\right) + 4 {\left(y - 0\right)}^{2} / \left(\frac{16}{3}\right) = 1$

${\left(x - \frac{2}{3}\right)}^{2} / \left(\frac{16}{9}\right) + {\left(y - 0\right)}^{2} / \left(\frac{16}{12}\right) = 1$

Write the denominators as squares:

${\left(x - \frac{2}{3}\right)}^{2} / {\left(\frac{4}{3}\right)}^{2} + {\left(y - 0\right)}^{2} / {\left(\frac{4}{\sqrt{12}}\right)}^{2} = 1$

The is standard Cartesian form of the equation of an ellipse with a center at $\left(\frac{2}{3} , 0\right)$; its semi-major axis is $\frac{4}{3}$ units long and is parallel to the x axis and its semi-minor axis is $\frac{4}{\sqrt{12}}$ units long and is parallel to the y axis.

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