Question

Find an equivalent equation in rectangular coordinates

Answer 1

`x^2+y^2 = 2x - 2y}`

Third answer option

Explanation:

We are given the polar equation as

`r = 2(\sin \theta - \cos \theta)`
and asked to convert it into rectangular form

We have the following equations which relate (r, θ) in polar form to (x, y) in rectangular form

`\cos \theta=(x)/(r) \rightarrow x=r \cos\theta\\\\\sin \theta=(y)/(r) \rightarrow y=r \sin \theta\\\\`

`r^2=x^2+y^2`

Original polar equation:

`r = 2(\sin \theta - \cos \theta)`

Expand the right side:

`r = 2\sin \theta - 2\cos \theta`

Substitute for sinθ and cosθ in terms of x and y

`r & = 2\left((y)/(r) - (x)/(r)\right)\\`

Multiply both sides by r:

`r^2 = 2(x - y)\\r^2 = 2x - 2y\\`

Substitute
`r^2=x^2+y^2` on the left side:

`\boxed{x^2+y^2 = 2x - 2y}`

This would be the third answer opton