# Identify the conic section given by the polar equation displaystyle{r}=frac{4}{{{1}- cos{theta}}} and also determine its directrix.

Identify the conic section given by the polar equation $r=\frac{4}{1-\mathrm{cos}\theta }$ and also determine its directrix.
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Step 1
We convert the equation into rectangular form first
$r=\frac{4}{1-\mathrm{cos}\theta }$
$r\left(1-\mathrm{cos}\theta \right)=4$
$r-r\mathrm{cos}\theta =4$
$\sqrt{{x}^{2}+{y}^{2}}-x=4$
$\sqrt{{x}^{2}+{y}^{2}}=x+4$
${x}^{2}+{y}^{2}={\left(x+4\right)}^{2}$
${x}^{2}+{y}^{2}={x}^{2}+8x+16$
${y}^{2}=8x+16$
${y}^{2}=8\left(x+2\right)$
This is a parabola.
Step 2
Compare the equation with the standard form
${\left(y-k\right)}^{2}=4p\left(x-h\right)$
$h=-2,k=0$
$8p=2$
$\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}},p=2$
Directrix $=x=h-p\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=-2-2\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=-4$
Answer: Parabola, $x=-4$
Jeffrey Jordon