find the equation of a circle given (-1,1) and (1 1/2, 5) are the endpoints of the diameter 

To find the equation of a circle, we need two key pieces of information: the center and the radius. Since the endpoints of the diameter are given, we can derive both from these points.

### Steps:
1. **Find the center of the circle**:
   The center of a circle is the midpoint of its diameter. The midpoint formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
   \[
   \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   Given the points \((-1, 1)\) and \((1 \frac{1}{2}, 5)\), we compute:
   \[
   \left( \frac{-1 + 1 \frac{1}{2}}{2}, \frac{1 + 5}{2} \right) = \left( \frac{-1 + \frac{3}{2}}{2}, \frac{6}{2} \right) = \left( \frac{\frac{1}{2}}{2}, 3 \right) = \left( \frac{1}{4}, 3 \right)
   \]
   So, the center is at \(\left( \frac{1}{4}, 3 \right)\).

2. **Find the radius**:
   The radius is half of the length of the diameter. To find the length of the diameter, we use the distance formula between the two endpoints:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Substituting \((-1, 1)\) and \((1 \frac{1}{2}, 5)\) into the formula:
   \[
   d = \sqrt{\left(1 \frac{1}{2} - (-1)\right)^2 + (5 - 1)^2} = \sqrt{\left(1 \frac{1}{2} + 1\right)^2 + 4^2} = \sqrt{\left(2 \frac{1}{2}\right)^2 + 16} = \sqrt{\left(\frac{5}{2}\right)^2 + 16}
   \]
   \[
   d = \sqrt{\frac{25}{4} + 16} = \sqrt{\frac{25}{4} + \frac{64}{4}} = \sqrt{\frac{89}{4}} = \frac{\sqrt{89}}{2}
   \]
   So, the diameter is \(\frac{\sqrt{89}}{2}\), and the radius \(r\) is half of this:
   \[
   r = \frac{\frac{\sqrt{89}}{2}}{2} = \frac{\sqrt{89}}{4}
   \]

3. **Write the equation of the circle**:
   The general equation of a circle with center \((h, k)\) and radius \(r\) is:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Substituting \(h = \frac{1}{4}, k = 3\), and \(r = \frac{\sqrt{89}}{4}\), we get:
   \[
   \left( x - \frac{1}{4} \right)^2 + (y - 3)^2 = \left( \frac{\sqrt{89}}{4} \right)^2
   \]
   Simplifying:
   \[
   \left( x - \frac{1}{4} \right)^2 + (y - 3)^2 = \frac{89}{16}
   \]

Thus, the equation of the circle is:
\[
\left( x - \frac{1}{4} \right)^2 + (y - 3)^2 = \frac{89}{16}
\]

Would you like any further details or clarifications?

### Related questions:
1. How do you derive the general equation of a circle from its center and radius?
2. What is the geometrical significance of the radius in a circle?
3. How can you find the center of a circle if only the equation is given?
4. What is the importance of the distance formula in geometry?
5. Can a circle be uniquely defined with just two points?

### Tip:
Always double-check the calculations involving midpoints and distances, especially with fractions, to avoid small errors!

Find the equation of a circle given (-1,1) and (1 1/2, 5) are the endpoints of the diameter.

Learn how to find the equation of a circle when given the endpoints of its diameter: (-1,1) and (1 1/2, 5). This solution involves using the midpoint and distance formulas to calculate the center and radius of the circle. Suitable for high school students studying geometry.

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