how do i find an equation for the hyperbola using what is given??  focus at (-4,0); vertices at (-4,4) and (-4,2) not sure where to start.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to start by remembering the equation of a hyperbola centered at `(h,k)`  such that:

`(x - h)^2/a^2- (y - k)^2/b^2 = 1`

You may start finding the coordinates of center of hyperbola since the problem provides the coordinates of vertices and the center of hyperbola is the midpoint of the segment joining the vertices of hyperbola, such that:

`h = (-4 - 4)/2 => h = -4`

`k = (4 + 2)/2 => k = 3`

Hence, you may substitute `-4`  for `h`  and `3`  for `k`  in equation of hyperbola such that:

`(x+4)^2/a^2 - (y - 3)^2/b^2 = 1`

Since the distance between any vertex and center of hyperbola gives the value of a, you may evaluate a using the distance formula such that:

`a = sqrt((-4 + 4)^2 + (4 - 3)^2) => a = 1`

You need to use the inmformation provided by the problem with regards to foci to evaluate b, such that:

`c^2 = a^2 + b^2`

Since c = -4 and a = 1 yields:

`16 = 1 + b^2 => b^2 = 15`

Since `b^2`  is determined, the equation of hyperbola is fully determined such that:

`(x+4)^2/1^2 - (y - 3)^2/(sqrt 15)^2 = 1`

Hence, evaluating the equation of hyperbola, under the given conditions, yields `(x+4)^2/1^2 - (y - 3)^2/(sqrt 15)^2 = 1` .

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