Write an equation for the hyperbola with vertices (0, - 1) and (0, 3) and foci (0, - 3) and (0, 5) .

Please see the explanation for the process. The equation is

y²/3²−x²/4²=1

Explanation:

There are two types of hyperbolas, one where a line drawn through its vertices and foci is horizontal, and one where a line drawn through its vertices and foci 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)²a²−(x−h)²b²=1

Observe that the x coordinate of the foci and the vertices is 0; this tells us that  

h=0(y−k)²a²−(x−0)²b²=1

The value of k is the sum of y coordinate of the vertices divided by two:

k=−3+32=0(y−0)²a²−(x−0)²b²=1

Please notice that when  

x=0,y=±3

; this allows us to find the value of "a":

(3−0)²a²−(0−0)²b²=1(3−0)²a²=1(3−0)²=(a²)a=3(y−0)²3²−(x−0)²b²=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:

c²=a²+b²5²=3²+b²b²=25−9b²=16b=4(y−0)²3²−(x−0)²4²=1

Simplifying a bit:

y²3²−x²4²=1

Step-by-step explanation:

click the image for best results

step-by-step explanation: