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{\bf Question}

An ellipse has polar equation $$\frac{2}{r} = 1 - \frac{1}{3} \cos
\theta.$$  Find the cartesian equation to axis having their origin
at the centre of the ellipse.

\vspace{.25in}

{\bf Answer}

$$\frac{2}{r} = 1 - \frac{1}{3} \cos \theta \hspace{.3in} {\rm So\
} l = 2 {\rm \ and \ }e = \frac{1}{3} $$ The cartesian equation is
$\ds \frac{x^2}{a^2} + \frac{y^2}{a^2(1-e^2)} = 1$

When $\ds x = \frac{1}{3}a$ and $\ds y = l (=2)$ then
$$\frac{1}{9} \frac{a^2}{a^2} +
\frac{4}{a^2\left(\frac{8}{9}\right)} = 1 \hspace{.3in}{\rm So\ }
a = \frac{9}{4} \, \, \, b^2 = \frac{9}{2}$$ So the equation is
$$16x^2 + 18y^2 = 81$$

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