This is a great opportunity to review the properties of hyperbolas! We've been given that the hyperbola has foci at points (2, 0) and (2, 6). We also know that the general form of a hyperbola is given by
( x - h )²/b² - ( y - k )²/a² = 1 for a hyperbola that extends along the x-axis or
( y - k )²/b² - ( x - h )²/a² = 1 for a hyperbola that extends along the y-axis
where (h, k) is the center of the hyperbola and 2a is the distance between foci.
We can figure out the parameters for our equation using the following steps:
Step 1: Calculate distance and center
From the given foci, we know that the distance between the foci is |6 - 0| = 6 units. This gives us 2a = 6, which means a = 3.
The center of the hyperbola can be found by averaging the coordinates of the foci. Thus, the center is at the point (h , k) = (2 , 3).
Step 2: Calculate value of b
The slopes of the asymptotes are ±1/b. From the given equations of the asymptotes, we can see the slopes are ±1/2. This tells us that b = 2.
Step 3: Write the equation
Now we have the values for a, b, and (h,k). We only need to insert these into the general form of the equation. Because the foci are vertically aligned, this hyperbola extends along the y-axis. Therefore, we need the form where ( y - k )² comes first.
So, the equation for the hyperbola is (y - 3)²/4 - (x - 2)²/9 = 1. This equation represents a hyperbola with foci at (2, 0) and (2, 6) and asymptotes defined by the lines y = 2 + 1/2 x and y = 4 - 1/2 x.