$\"\"$

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The vertices are $\"\"$ and point is $\"\"$ .

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Observe vertices and foci, $\"\"$ coordinates are equal.

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So, the hyperbola has a vertical transverse axis and its standard form is

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$\"\"$ .

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Where, $\"\"$ is the center.

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The distance between center and vertex is $\"\"$ .

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The distance between center and focus is $\"\"$ .

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$\"\"$ .

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Using midpoint formula, the center of the hyperbola is

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$\"\"$

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The distance between center and vertex is $\"\"$ .

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$\"\"$

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Find $\"\"$ .

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The hyperbola $\"\"$ is passing through the point $\"\"$ .

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Substitute $\"\"$ , $\"\"$ , and $\"\"$ in $\"\"$ .

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$\"\"$ \ \

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Substitute $\"\"$ , $\"\"$ , and $\"\"$ in $\"\"$ .

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$\"\"$

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The hyperbola equation is $\"\"$ .

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$\"\"$

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The hyperbola equation is $\"\"$ .