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

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Answer 1 · 2017-01-27T06:45:18

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

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 · 2017-01-27T06:46:30

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

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.