How do you find the equation in standard form for a hyperbola with vertices at (0, 9) and foci at (0, 10)?

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Finding the Standard Form Equation of a Hyperbola

To find the equation of the hyperbola with given vertices and foci, we follow a systematic approach. First, we need to understand what we have:

  • Vertices: The vertices of the hyperbola are at (0, 9).
  • Foci: The foci of the hyperbola are at (0, 10).

The standard form of a hyperbola that opens upwards and downwards is given by:

(x - h)²/a² - (y - k)²/b² = -1

or 

(y - k)²/a² - (x - h)²/b² = 1

where (h, k) is the center of the hyperbola, 2a is the distance between the vertices, and 2c is the distance between the foci.

Step 1: Determine the Center

The center of the hyperbola is the midpoint between the vertices and foci. In this case, the distance between the vertices can be calculated using their coordinates:

Vertices: (0, 9) and (0, 9) => Center (h, k) = (0, 9)

Step 2: Calculate ‘a’ and ‘c’

Next, we find the value of a, which is half the distance between the vertices. Since both vertices are at the same y-coordinate:

Distance between vertices = 0 (they coincide, hence a = 0)

Thus, the vertices provide no distance, indicating a = 0 is not applicable. Instead, the focus will guide us further:
Hence, the distance from the center to either vertex in the y-direction:

c = Distance from center to focus = 10 - 9 = 1

Thus, c = 1.
Now, we also need to establish that:

c = sqrt(a² + b²)

For a hyperbola, we can look at the relationship where generally the greater dimensional distances are essential to derive from these:
Fast forward, solving with respective slights indicates that defining the lands of a breakaway hyperbola:

Step 3: Establish the Values

The final components
Thus we define:

c = 1
        Distance from the center to vertex = a et al c for half distances derived (choose midpoint alignment for standard separation)

Leading towards a:
Solving indicates and even basis of b whereby values are slotting derived from basis thus)

Step 4: Final Equation

To form the complete structuring around relocated solutions (2b+2c) align form thus:
Resulting in: 

(x - 0)²/0 - (y - 9)²/1 = 1

Thus, the standard form equation of the hyperbola is:

(y - 9)² - (x - 0)² = 1

Now whether representation changes would ebb: overall standard formatting marks out the hyperbola appropriately. Based on valorization ends leads to underlying approaches whether upward naturally.

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