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## Homework Statement

The equation of a conic in polar coordinates is:

[tex] r = \frac{r_o}{1-\epsilon cos(\theta)}[/tex].

[tex]\epsilon [/tex] is the eccentricity, 0 for a circle, (0,1) for an ellipse, 1 for a parabola, and >1 for a hyperbola.

What is this equation expressed in Cartesian coordinates?

## Homework Equations

[tex]r=(x^2+y^2)^{1/2}[/tex]

[tex]x=r cos(\theta)[/tex]

## The Attempt at a Solution

[tex] r = \frac{r_o}{1-\epsilon cos(\theta)}[/tex]

[tex] r (1-\epsilon cos(\theta)) = r_o[/tex]

[tex] r - r cos(\theta) \epsilon = r_o[/tex]

[tex] r - x \epsilon = r_o[/tex]

[tex] r = r_o + x \epsilon[/tex]

[tex] (r)^2 = (r_o + x \epsilon)^2 [/tex]

[tex] x^2 + y^2 = r_o^2 + 2 (x \epsilon) r_o + (x \epsilon)^2 [/tex]

[tex] x^2 - (x \epsilon)^2 - 2 \epsilon r_o x + y^2 = r_o^2 [/tex]

[tex] (1 - \epsilon^2)x^2 - 2 \epsilon r_o x + y^2 = r_o^2 [/tex]

My textbook says this equation should be:

[tex] (1 - \epsilon^2)x^2 - 2 r_o x + y^2 = r_o^2 [/tex]

(Notice that we differ on the coefficient of x. Is the textbook missing an epsilon by a misprint? or did I mess up somewhere?

Thanks for the help.