$\"\"$

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The equation is $\"\"$ .

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Rewrite the above equation.

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$\"\"$

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Compare the above 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 vertices are at the end points of the major axis.

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The vertices occur when $\"\"$ and $\"\"$ .

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Case 1 : When $\"\"$ .

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$\"\"$

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

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$\"\"$

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$\"\"$

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Case 2 : When $\"\"$ .

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$\"\"$

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

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$\"\"$

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The polar coordinates of the vertices are $\"\"$ and $\"\"$ correspond to the rectangular coordinates are $\"\"$ and $\"\"$ .

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The ellipse’s center is the midpoint of the segment between 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 cooresponding values in $\"\"$ .

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$\"\"$

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$\"\"$ .

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

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$\"\"$

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