The points (5,2) and (−3,8) are at opposite ends of the diameter of a circle. Determine the equation of the circle.
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To determine the equation of the circle, we need to follow these steps:

1. **Find the center of the circle**:  
   The center of the circle is the midpoint of the diameter. The coordinates of the midpoint \( C(x_c, y_c) \) can be found using the midpoint formula:

   \[
   C(x_c, y_c) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   For the points \( (5, 2) \) and \( (-3, 8) \):

   \[
   C(x_c, y_c) = \left( \frac{5 + (-3)}{2}, \frac{2 + 8}{2} \right) = \left( \frac{2}{2}, \frac{10}{2} \right) = (1, 5)
   \]

   So, the center of the circle is \( (1, 5) \).

2. **Find the radius of the circle**:  
   The radius is half the length of the diameter, and the length of the diameter is the distance between the two given points. We use the distance formula to calculate the length of the diameter:

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

   Substituting the values of the points \( (5, 2) \) and \( (-3, 8) \):

   \[
   d = \sqrt{((-3) - 5)^2 + (8 - 2)^2} = \sqrt{(-8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
   \]

   Since the radius \( r \) is half of the diameter, we have:

   \[
   r = \frac{d}{2} = \frac{10}{2} = 5
   \]

3. **Write the equation of the circle**:  
   The general equation of a circle is:

   \[
   (x - x_c)^2 + (y - y_c)^2 = r^2
   \]

   Substituting the values of the center \( (1, 5) \) and the radius \( 5 \):

   \[
   (x - 1)^2 + (y - 5)^2 = 5^2
   \]
   \[
   (x - 1)^2 + (y - 5)^2 = 25
   \]

Thus, the equation of the circle is:

\[
(x - 1)^2 + (y - 5)^2 = 25
\]

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Would you like more details or have any questions about this solution?

Here are five related questions to explore further:

1. How would you find the equation of a circle if given a different pair of diameter endpoints?
2. What is the distance formula, and how is it derived?
3. How can you determine the equation of a circle if given its center and another point on the circle?
4. What is the general form of a circle equation, and how can it be converted from the standard form?
5. How can the concept of midpoints be used in other geometric problems?

**Tip**: To verify a circle's equation, plug in the coordinates of any point that lies on the circle into the equation. If it satisfies the equation, the point is indeed on the circle.

The points (5,2) and (−3,8) are at opposite ends of the diameter of a circle. Determine the equation of the circle.

Learn how to determine the equation of a circle when given two points at opposite ends of the diameter, specifically (5, 2) and (−3, 8). This problem involves key geometric concepts such as the midpoint and distance formulas, and provides a detailed step-by-step solution. Suitable for students in Grades 9-11.

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