The coordinates of the points a and b are (-1,-2) and (7,4) respectively. Find the equation of the circle, c, for which ab is the diameter.

To find the equation of the circle \( C \) with diameter \( AB \), we need to follow these steps:

1. **Calculate the midpoint of \( A \) and \( B \)**:
   The midpoint of the diameter will be the center of the circle. If the coordinates of points \( A \) and \( B \) are \( (-1, -2) \) and \( (7, 4) \) respectively, the midpoint \( M \) can be calculated as:
   \[
   M = \left( \frac{-1 + 7}{2}, \frac{-2 + 4}{2} \right) = \left( \frac{6}{2}, \frac{2}{2} \right) = (3, 1)
   \]
   So, the center of the circle \( C \) is at \( (3, 1) \).

2. **Calculate the radius of the circle**:
   The radius is half the length of the diameter \( AB \). The distance \( AB \) can be found using the distance formula:
   \[
   AB = \sqrt{(7 - (-1))^2 + (4 - (-2))^2} = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
   \]
   Since \( AB \) is the diameter, the radius \( r \) is:
   \[
   r = \frac{10}{2} = 5
   \]

3. **Write the equation of the circle**:
   The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Here, \( h = 3 \), \( k = 1 \), and \( r = 5 \). Substituting these values, we get:
   \[
   (x - 3)^2 + (y - 1)^2 = 5^2
   \]
   \[
   (x - 3)^2 + (y - 1)^2 = 25
   \]

### Final Answer
The equation of the circle \( C \) is:
\[
(x - 3)^2 + (y - 1)^2 = 25
\]

Would you like a further breakdown of any part?

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**Related Questions:**
1. How do you find the midpoint of a line segment between two points?
2. What are the steps to find the length of a line segment given two endpoints?
3. How do you derive the equation of a circle given the center and radius?
4. How can the circle equation be rearranged to a general form \( Ax^2 + Ay^2 + Dx + Ey + F = 0 \)?
5. How does changing the diameter endpoints affect the circle’s equation?

**Tip:** To verify a circle’s equation, substitute known points to see if they satisfy the equation.

This problem demonstrates how to find the equation of a circle given two points, A (-1,-2) and B (7,4), as endpoints of the diameter. Learn step-by-step methods to find the center, radius, and final equation of the circle in standard form. Suitable for students in Grades 9-11.

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