Find a polar equation for the conic section. Assume one focus is at the origin.

Question

broliY · Accepted Answer

We will first consider the given graph,
The objective is to find the polar equation of the graph.
We are given that $y = - 2$ is the directrix nearer to the focus $(0, 0)$ ,
Thus, we have, $d$ $= 2$
Then the equation of the given ellipse will be in the form $r = \frac{2 e}{1 - e \sin θ}$
Now, from the figure, we will evaluate the length of the major axis
That is
$2 a =$ $\frac{8}{3}$
$a =$ $\frac{4}{3}$
Thus we get center is $(0, - \frac{8}{3})$
Next, we will find the distance from the center to the focus,
That is,
$a e =$ $\frac{8}{3}$
$\frac{4}{3} e = \frac{8}{3}$
$e = 2$
Thus the required equation of the elipse in polar form is as follows
$r =$ $\frac{4}{1 - 2 \sin θ}$

Find a polar equation for the conic section. Assume one focus is at the origin.

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