Step 1 :

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Identify the conic from its general equation :

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The graph of $\"image\"$ is one of the following

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1. Circle : $\"image\"$

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2. Parabola : $\"image\"$ , (either $\"image\"$ or $\"image\"$ but not both).

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3. Ellipse : $\"image\"$ , ( A and C have like signs).

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4. Hyperbola : $\"image\"$ , ( A and C have unlike signs).

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Step 2 :

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The equation is $\"\"$ .

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Rewrite the equation : $\"\"$

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Compare $\"\"$ with the general equation $\"image\"$ .

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$\"\"$

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$\"\"$ and A,C are having like signs.

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The graph of the equation represents an ellipse.

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Step 3 :

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The ellipse equation is $\"\"$ .

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Rewrite the equation into standard form of ellipse :

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$\"\"$

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Compare it to standard form of vertical ellipse is $\"\"$ .

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Where $\"image\"$

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a is length of semi major axis and b is length of semi minor axis.

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Center is $\"\"$ , vertices $\"\"$ .

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Foci $\"\"$ .

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Where $\"\"$ .

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$\"\"$

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Now compute the c :

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$\"\"$

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vertices :

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$\"\"$

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Foci :

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$\"\"$

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Solution :

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(a) The graph of the equation represents an ellipse.

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(b) Vertices $\"\"$ and Foci $\"\"$ .