How to write the equation of the hyperbola given Foci: (0,-5),(0, 5) and vertices (0, -3), (0,3)?

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Julissa Heath · Accepted Answer

There are two different kinds of hyperbolas: one where the line connecting its vertices and foci is horizontal, and the other where the line is vertical. This hyperbola is the type where a line drawn through its vertices and foci is vertical. We know this by observing that it is the y coordinate that changes when we move from a focus point to a vertex. The general equation for this type of hyperbola is:                              (          y          -          k          )                ²                    a        ²              -                            (          x          -          h          )                ²                    b        ²              =    1  Observe that the x coordinate of the foci and the vertices is 0; this tells us that       h    =    0                                (          y          -          k          )                ²                    a        ²              -                            (          x          -          0          )                ²                    b        ²              =    1  The value of k is the sum of y coordinate of the vertices divided by two:      k    =                  -        3        +        3            2        =    0                                (          y          -          0          )                ²                    a        ²              -                            (          x          -          0          )                ²                    b        ²              =    1  Please notice that when       x    =    0    ,    y    =    ±    3  ; this allows us to find the value of &quot;&quot;a&quot;&quot;:                              (          3          -          0          )                ²                    a        ²              -                            (          0          -          0          )                ²                    b        ²              =    1                                (          3          -          0          )                ²                    a        ²              =    1              (      3      -      0      )        ²    =          (      a      ²      )            a    =    3                                (          y          -          0          )                ²                    3        ²              -                            (          x          -          0          )                ²                    b        ²              =    1  Let c = the distance between the the center point and focus = 5. The equation for the square of this distance helps us to find the value of b:      c    ²    =    a    ²    +    b    ²        5    ²    =    3    ²    +    b    ²        b    ²    =    25    -    9        b    ²    =    16        b    =    4                                (          y          -          0          )                ²                    3        ²              -                            (          x          -          0          )                ²                    4        ²              =    1  Simplifying a bit:                    y        ²                    3        ²              -                  x        ²                    4        ²              =    1

How to write the equation of the hyperbola given Foci: (0,-5),(0, 5) and vertices (0, -3), (0,3)?

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