Question

Transform the following polar equation into an equation in rectangular coordinates: = -4 sine O A. x+y = -4 ов. у = -4 Oc+(y-2 - 4 OD. x = -44

Answer 1

The relation between cartesian coordinates (x, y) and polar coordinates (r, θ), is given by the following equations:

`\begin{gathered} x=r\cdot\cos \theta, \\ y=r\cdot\sin \theta, \\ r^2=x^2+y^2. \end{gathered}`

We must convert the following equation from polar coordinates to cartesian coordinates:

`r=-4\cdot\sin \theta.`

1) We multiply both sides of the equation by r:

`\begin{gathered} r\cdot r=r\cdot(-4\cdot\sin \theta)\text{.} \\ r^2=-4\cdot(r\cdot\sin \theta). \end{gathered}`

2) Now, we replace by the identities above:

`x^2+y^2=-4\cdot y.`

3) We rewrite the equation in the following way:

`\begin{gathered} x^2+y^2+4y=0, \\ x^2+(y^2+2\cdot2y+4)-4=0, \\ x^2+(y+2)^2-4=0, \\ x^2+(y+2)^2=4. \end{gathered}`

Answer

C.

`x^2+(y+2)^2=4`