We are given a hyperbola $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$, and the task is to find the formula for the distance from any point $(x, y)$ on the hyperbola, where $y > 0$, to the point $\left( 0, \sqrt{a^2 + b^2} \right)$.

Steps:

1. Distance formula: The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane is given by:

   $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

   In this case, the distance from $(x, y)$ on the hyperbola to $\left( 0, \sqrt{a^2 + b^2} \right)$ is:

   $d = \sqrt{(x - 0)^2 + \left( y - \sqrt{a^2 + b^2} \right)^2}$

   Simplifying, we have:

   $d = \sqrt{x^2 + \left( y - \sqrt{a^2 + b^2} \right)^2}$

2. Relation between $x$ and $y$ on the hyperbola: Since $(x, y)$ lies on the hyperbola, the equation $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ holds. From this, we can express $x^2$ in terms of $y^2$:

   $\frac{x^2}{a^2} = \frac{y^2}{b^2} - 1$

   Solving for $x^2$:

   $x^2 = a^2 \left( \frac{y^2}{b^2} - 1 \right)$

   $x^2 = \frac{a^2 y^2}{b^2} - a^2$

3. Substitute into the distance formula: Now we substitute $x^2$ into the distance formula:

   $d = \sqrt{\left( \frac{a^2 y^2}{b^2} - a^2 \right) + \left( y - \sqrt{a^2 + b^2} \right)^2}$

   This gives the required distance formula in terms of $y$.

Would you like further details or assistance with any step in the process? Also, here are five related questions that can extend the topic:

1. What are the asymptotes of the given hyperbola?
2. How do we derive the equation of a hyperbola from its geometric definition?
3. What is the geometric meaning of the distance calculated in this problem?
4. How does the shape of the hyperbola change with different values of $a$ and $b$?
5. Can we generalize this problem for hyperbolas that are not centered at the origin?

Tip: When working with conic sections, always try to simplify one variable in terms of the other using the defining equation before applying more complex formulas.