Question

How do you change the polar equation `r(1+costheta)=1` into rectangular form?

Answer 1

The equation is `y^2=1-2x`

Explanation:

To change from polar coordinates `(r,theta)` to rectangular coordinates, we use the following equations

`x=rcostheta`

`y=rsintheta`

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

Therefore,

`r(1+costheta)=1`

`r+rcostheta=1`

`sqrt(x^2+y^2)+x=1`

`sqrt(x^2+y^2)=1-x`

Squaring both sides

`cancelx^2+y^2=1-2x+cancelx^2`

So,

`y^2=1-2x`

Answer 2

`"The desired Cartesian Eqn. is "y^2+2x-1=0.`

Explanation:

The Formula for converting polar to rectangular form, are as under :

`x=rcostheta, y=rsintheta, and, x^2+y^2=r^2.............(star)`

`"Now, "r(1+costheta)=1 rArr r+rcostheta=1`

Using `(star)`, the eqn. becomes

`sqrt(x^2+y^2)+x=1 rArr {sqrt(x^2+y^2)}^2=(1-x)^2`

`rArr x^2+y^2=1-2x+x^2`

`:. y^2+2x-1=0,` is the desired Cartesian Eqn.