$\"\"$ \ \

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The equation is $\"\"$ . \ \

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Compare the equation with polar equation of conic $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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Since $\"\"$ , the equation is ellipse.

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The directrix of the ellipse $\"\"$ .

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The general equation of ellipse in rectangular form : $\"\"$ .

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The directrix of the ellipse occur when $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$

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$\"\"$

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The vertex lies at polar coordinate $\"\"$ correspond to rectangular coordinates $\"\"$ .

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Substitute $\"\"$ in the above equation.

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$\"\"$

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The vertex lies at polar coordinate $\"\"$ correspond to rectangular coordinates $\"\"$ .

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The ellipse center is the midpoint of the vertices $\"\"$

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The distance between the center and each vertex is $\"\"$ .

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The distance from the center $\"\"$ to the focus at $\"\"$ is $\"\"$ .

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$\"\"$ \ \

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$\"\"$

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Substitute $\"\"$ and $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$ .

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Therefore, the equation of the ellipse is $\"\"$ .

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$\"\"$

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The equation of the ellipse is $\"\"$ .