
In the xy-plane, a circle has center C with coordinates (h, k). Points A and B lie on the circle. Point A has coordinates (h + 1, k + √102), and ZACB is a right angle. What is the length of AB?
A) √206
B) 2√102
C) 103√2
D) 103√3

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.

In the xy-plane, a circle has center C with coordinates (h, k). Points A and B lie on the circle. Point A has coordinates (h + 1, k + √102), and ZACB is a right angle. What is the length of AB?
A) √206
B) 2√102
C) 103√2
D) 103√3

Explore a detailed solution to a circle geometry problem involving a right angle and chord length. Learn how to calculate the distance between points on a circle using symmetry and key theorems like the Cyclic Quadrilateral and Inscribed Angle Theorems. Suitable for Grades 10-12.

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