find an equation of the circle whose diameter has endpoints (-3,3) and (5,6)

To find the equation of the circle given the diameter endpoints \((-3, 3)\) and \((5, 6)\), we follow these steps:

1. **Find the center of the circle:**
   The center of the circle is the midpoint of the diameter. The midpoint \((x_c, y_c)\) of two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
   \[
   \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   Substituting the given points \((-3, 3)\) and \((5, 6)\):
   \[
   x_c = \frac{-3 + 5}{2} = \frac{2}{2} = 1
   \]
   \[
   y_c = \frac{3 + 6}{2} = \frac{9}{2} = 4.5
   \]
   So, the center of the circle is \((1, 4.5)\).

2. **Find the radius of the circle:**
   The radius \(r\) is half the distance between the diameter's endpoints. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Substituting the given points:
   \[
   d = \sqrt{(5 - (-3))^2 + (6 - 3)^2} = \sqrt{(5 + 3)^2 + (6 - 3)^2} = \sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73}
   \]
   The radius \(r\) is half of \(d\):
   \[
   r = \frac{\sqrt{73}}{2}
   \]

3. **Form the equation of the circle:**
   The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Substituting the center \((1, 4.5)\) and the radius \(\frac{\sqrt{73}}{2}\):
   \[
   (x - 1)^2 + (y - 4.5)^2 = \left( \frac{\sqrt{73}}{2} \right)^2
   \]
   Simplifying the right-hand side:
   \[
   \left( \frac{\sqrt{73}}{2} \right)^2 = \frac{73}{4}
   \]
   Therefore, the equation of the circle is:
   \[
   (x - 1)^2 + (y - 4.5)^2 = \frac{73}{4}
   \]

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the distance between two points in a coordinate plane?
2. What is the midpoint formula and how is it derived?
3. How do you derive the equation of a circle from its geometric definition?
4. How do you convert the general form of a circle's equation to the standard form?
5. How do you find the radius of a circle given its equation in standard form?
6. Can you explain the significance of the center and radius in the equation of a circle?
7. How do you graph a circle given its equation?
8. What is the distance formula and how is it used in geometry?

**Tip:** Always verify your calculations by plugging the coordinates back into the equation to ensure the points satisfy the circle's equation.

Learn how to find the equation of a circle given the endpoints of its diameter (-3, 3) and (5, 6). Step-by-step solution includes finding the circle's center, radius, and forming the equation in standard form. Suitable for high school students.

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