Question

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

Answer 1

`{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`