Question

Find an equivalent equation in rectangular coordinates.r(1 - 2 cos θ) = 1

Answer 1

The given equation is:

`r(1-2\cos\theta)=1`

Rewrite the equation

`r-2r\cos\theta=1`

Recall that

`r=√(x^2+y^2)x=rcos\theta`

Substituting these values gives

`\begin{gathered} √(x^2+y^2)-2x=1 \\ (√(x^2+y^2))^2=(1+2x)^2 \\ x^2+y^2=1+4x^2+4x \\ 1+4x^2+4x-x^2-y^2=0 \\ 3x^2-y^2+4x+1=0 \end{gathered}`

Therefore the required equation is

`3x^(2)-y^(2)+4x+1=0`