Can it be shown that gives a conic section?

$r}^{2}=\frac{1}{1-\epsilon \mathrm{cos}\left(2\theta \right)$

Joey Rodgers
2022-03-18
Answered

Can it be shown that gives a conic section?

$r}^{2}=\frac{1}{1-\epsilon \mathrm{cos}\left(2\theta \right)$

You can still ask an expert for help

PCCNQN4XKhjx

Answered 2022-03-19
Author has **8** answers

Step 1

Rewrite it as

${r}^{2}(1-\u03f5\mathrm{cos}\left(2\theta \right))$

$=1\iff {r}^{2}(1-\u03f5({\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta ))=1$

and then expand. We have

$r}^{2}={x}^{2}+{y}^{2},\text{}{r}^{2}{\mathrm{cos}}^{2}\theta ={x}^{2$

and

$r}^{2}{\mathrm{sin}}^{2}\theta ={y}^{2$

so we get the traditional equation of an ellipse or a hyperbola.

Rewrite it as

and then expand. We have

and

so we get the traditional equation of an ellipse or a hyperbola.

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