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

Choices:

r4 = -4r2

r2 = 4cos2θ

r2 = 4sin2θ

User Kirbo
by
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1 Answer

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ANSWER


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

Step-by-step explanation

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

User Mangusbrother
by
8.7k points

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