Question

Standard form of equation of hyperbola with given characteristics:
A) vertices (1,2) and (1,-2) and passes through the point (0,√5)

Answer 1

The center is directly between the two vertices, meaning that the center is at 1,0

Answer 2

Because the two vertices are on the same vertical line, the hyperbola is vertical, meaning that the equation is a variation of \[(y-0)^2 - (x-1)^2 = 1\].

Using the vertices once more, we see that the length of the major axis of the hyperbola must have length 4. This means that the radius has length 2, and 2^2 = 4.

\[\frac{ y^2 }{ 4 } - \frac{ (x-1)^2 }{ b^2 } = 1\]

Plugging in 0,sqrt 5 into the equation, we have that\[\frac{1}{b^2} = \frac{1}{4}\]
so b^2 is 4.

Our final answer is:

\\[\frac{ y^2 }{ 4 } - \frac{ (x-1)^2 }{ 4 } = 1\]

Answer 3

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