$\"\"$

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The general form of equation is $\"\"$ .

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The rotation equation is $\"\"$ .

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From exercise (55): $\"\"$ , hence the value of $\"\"$ is invariant.

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Consider $\"\"$ so that $\"\"$ .

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Then, $\"\"$ .

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(a)

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If $\"\"$ then $\"\"$ .

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If $\"\"$ , then either $\"\"$ or $\"\"$ , but not both, so the form of equation either $\"\"$ or $\"\"$ .

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Consider the equation $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The vertex of the parabola is $\"\"$ and the axis of symmetry is parallel to $\"\"$ -axis.

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$\"\"$

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Consider the equation $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The vertex of the parabola is $\"\"$ and the axis of symmetry is parallel to $\"\"$ -axis.

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Therefore, $\"\"$ then the conic represents parabola.

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$\"\"$

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(b) If $\"\"$ then $\"\"$ .

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If $\"\"$ , then $\"\"$ and $\"\"$ are of the same sign.

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

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$\"\"$

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$\"\"$

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$\"\"$

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Completing the squares by adding $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Let $\"\"$ .

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$\"\"$ .

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If $\"\"$ .

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$\"\"$

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$\"\"$

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Here $\"\"$ and $\"\"$ are same signs.

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Therefore, the equation $\"\"$ represents an ellipse. $\"\"$

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(c) If $\"\"$ then $\"\"$ .

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If $\"\"$ , then $\"\"$ and $\"\"$ are of the opposite sign.

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

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$\"\"$

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$\"\"$

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$\"\"$

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Completing the squares by adding $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Let $\"\"$ .

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$\"\"$ .

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If $\"\"$ .

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$\"\"$

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$\"\"$

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Here $\"\"$ and $\"\"$ are opposite signs.

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Therefore, the equation $\"\"$ represents a hyperbola.

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$\"\"$

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(a) $\"\"$ represents a parabola.

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(b) $\"\"$ represents an ellipse.

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(c) $\"\"$ represents a hyperbola.