Given the problem setup:

• A circle has center $C(h, k)$.
• Point $A$ is on the circle at $(h + 1, k + \sqrt{102})$.
• There is a right angle at $ZACB$, meaning $\angle ZACB = 90^\circ$.

Step 1: Coordinates of Point A and C

From the given, the center of the circle is at $C(h, k)$ and point $A$ is at $(h+1, k+\sqrt{102})$. This implies that $A$ is shifted by 1 unit horizontally and by $\sqrt{102}$ units vertically from $C$.

Step 2: Understanding $ZACB$ as a Right Angle

The condition $\angle ZACB = 90^\circ$ suggests that the points $Z, A, C, B$ form a cyclic quadrilateral, and the right angle occurs between certain segments involving these points.

However, we are primarily tasked with finding the distance $AB$, and this condition implies symmetry in the points on the circle.

Step 3: Symmetry of the Problem

Since point $A$ is symmetrically placed with respect to the circle and we are dealing with a right angle, it's reasonable to consider that point $B$ might be symmetrically placed with respect to $C$ as well.

Step 4: Distance Between Points on a Circle

The key here is to determine the length of the chord $AB$. Given the symmetry and that the right angle divides the circle into predictable segments, the correct distance would be a simple factor of the radius of the circle.

Answer:

The length of $AB$ is $\boxed{2\sqrt{102}}$. Therefore, the correct answer is:

B) $2\sqrt{102}$.

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Would you like to go through more detailed steps or clarify any part of the problem? Here are some additional related questions for practice:

1. How do you derive the equation of a circle given points on it?
2. Can you find the radius of the circle given the center and a point on the circumference?
3. What is the geometric significance of cyclic quadrilaterals?
4. How does symmetry help in solving circle-related problems?
5. How do you calculate the distance between two points on a coordinate plane?

Tip: Remember that right angles inscribed in circles often suggest cyclic quadrilaterals, which can simplify calculations.