# The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:displaystyle{e}={2}, directrix displaystyle{r}=-{2} sec{theta}.

The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:
$e=2,$
$r=-2\mathrm{sec}\theta .$

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Neelam Wainwright
$e=2,r=-2\mathrm{sec}\theta$
Since $e>1$, therefore the conic is a hyperbola.
Now consider the directrix,
$r=-2\mathrm{sec}\theta$
$⇒r=-\frac{2}{\mathrm{cos}\theta }$
$⇒r\mathrm{cos}\theta =-2$
$⇒x=-2$
Now comparing whith $x=-p,$ we get
$p=2$
Therefore equation is,
$r=\frac{ep}{1-e\mathrm{cos}\theta }$
$⇒r=\frac{2\cdot 2}{1-2\mathrm{cos}\theta }$
$⇒r=\frac{4}{1-2\mathrm{cos}\theta }$
Now the graph is,