How do you convert $r = \frac{1}{1 - cos (θ)}$ $r = 1/(1-cos(theta))$ into cartesian form?

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Answer 1 · 2016-08-09T14:34:27

$y^{2} = 2 x + 1$ $y^2=2x+1$ representing the parabola with axis ax =-1/2 and focus at the origin.

The conversion formula is $(x, y) = (r cos θ, r sin θ)$ $(x, y) = (rcos theta, rsin theta)$ .

The given equation is $r = \sqrt{x^{2} + y^{2}} = \frac{1}{1 - \frac{x}{\sqrt{x^{2} + y^{2}}}}$ $r = sqrt (x^2+y^2)=1/(1-x/sqrt(x^2+y^2)$

Cross multiplying, rationalizing and simplifying,

$y^{2} = 2 x + 1$ $y^2=2x+1$

${(y - β)}^{2} = 4 a (s - α)$ $(y-beta)^2=4a(s-alpha)$ ,

representing parabolas having vertex at #(alpha, beta) parameter a

and focus at( $α + a, β)$ $alpha +a, beta)$ .

Here, $a = \frac{1}{2}, α = - \frac{1}{2} and β = 0$ $a = 1/2, alpha = -1/2 and beta = 0$ ...

How do you convert r = 1/(1-cos(theta))r=11−cos(θ)r = 1/(1-cos(theta)) into cartesian form?