$\"\"$

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The eccentricity of a conic is $\"\"$ .

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Vertices of the conic is at $\"\"$ and $\"\"$ .

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Because eccentricity is $\"\"$ , the conic is an ellipse.

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The center of the ellipse is at $\"\"$ , the midpoint of the segment between the given vertices.

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Therefore the directrix will be in the left side of the pole i.e. at $\"\"$ .

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The polar equation of the conic with the directrix is $\"\"$ .

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$\"\"$

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Find the value of $\"\"$ :

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Use the value of $\"\"$ and the polar form of a point on the conic to determine the value of $\"\"$ .

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The vertex $\"\"$ has polar coordinates $\"\"$ .

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$\"\"$ .

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$\"\"$ .

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$\"\"$ .

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The equation in standard form : $\"\"$ .

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Polar form of the conic with directrix is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in standard form.

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$\"\"$

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

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By simplifying $\"\"$ .

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Therefore, the polar equation is $\"\"$ .

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Because $\"\"$ , the equation of the directrix is $\"\"$ .

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$\"\"$

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(1) Draw the coordinate plane.

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(2) Graph the polar equation $\"\"$ .

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Graph :

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$\"\"$

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Observe the graph,

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The polar equation is an ellipse.

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$\"\"$

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The polar equation is $\"\"$ .

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The polar equation is an ellipse.

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Graph :

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$\"\"$