Question

Convert the polar equation r^2 = 2sin2Ө to a Cartesian equation. Answer choices: (x^2 + y^2)^2 = 2xy (x^2 + y^2)^2 = 4xy (x^2 + y^2)^2 = 2y^2

Answer 1

(X^2+y^2)^2 =4xy

Explanation:

Answer 2

Start by using the substitution
`\sin 2\theta = 2 \sin \theta \cos \theta`:


`r^2 = 2 \sin 2\theta`

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

`r^2 = 4 \sin \theta \cos \theta`

Then, multiply both sides by
`r^2`:


`r^4 = 4r^2 \sin \theta \cos \theta`

`(r^2)^2 = 4(r \sin \theta)(r \cos \theta)`

Since
`r^2 = x^2 + y^2`,
`r \cos \theta = x`, and
`r \sin \theta = y`, we have that


`\bf (x^2 + y^2)^2 = 4xy`