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leva [86]
3 years ago
6

Convert the Cartesian equation (x 2 + y 2)2 = 4(x 2 - y 2) to a polar equation.

Mathematics
1 answer:
SashulF [63]3 years ago
7 0

<u>ANSWER</u>

{r}^{2}  = 4  \cos2\theta

<u>EXPLANATION</u>

The Cartesian equation is

{( {x}^{2}  +  {y}^{2} )}^{2}  = 4( {x}^{2} -  {y}^{2}  )

We substitute

x = r \cos( \theta)

y = r \sin( \theta)

and

{x}^{2}  +  {y}^{2}  =  {r}^{2}

This implies that

{( {r}^{2} )}^{2}  = 4(( { r \cos\theta)  }^{2} -  {(r \sin\theta) }^{2}  )

Let us evaluate the exponents to get:

{r}^{4}  = 4({  {r}^{2} \cos^{2}\theta } -   {r}^{2}  \sin^{2}\theta)

Factor the RHS to get:

{r}^{4}  = 4{r}^{2} ({   \cos^{2}\theta } -   \sin^{2}\theta)

Divide through by r²

{r}^{2}  = 4 ({   \cos^{2}\theta } -   \sin^{2}\theta)

Apply the double angle identity

\cos^{2}\theta -\sin^{2}\theta=  \cos(2 \theta)

The polar equation then becomes:

{r}^{2}  = 4  \cos2\theta

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