$\"\"$

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The equation is $\"\"$ .

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Rewrite the above equation $\"\"$ .

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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 hyperbola.

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The directrix of the hyperbola is $\"\"$ .

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

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The vertices lie on the transverse 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 hyperbola’s 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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Substitute corresponding values in the general equation of hyperbola.

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$\"\"$

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$\"\"$

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Substitute corresponding values in $\"\"$ .

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$\"\"$

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

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$\"\"$

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