Give the standard equation of the circle with a diameter with endpoints A(-1, 4) and B(4, 2)

To find the standard equation of a circle given the endpoints of its diameter, we can follow these steps:

### Step 1: Find the center of the circle
The center of the circle is the midpoint of the diameter. To find the midpoint \(M(x, y)\) of two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), we use the midpoint formula:

\[
M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

For points \(A(-1, 4)\) and \(B(4, 2)\):

\[
M\left(x, y\right) = \left(\frac{-1 + 4}{2}, \frac{4 + 2}{2}\right) = \left(\frac{3}{2}, 3\right)
\]

So, the center of the circle is \( \left(\frac{3}{2}, 3\right) \).

### Step 2: Find the radius of the circle
The radius is half the length of the diameter. We can find the length of the diameter using the distance formula:

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

For points \(A(-1, 4)\) and \(B(4, 2)\):

\[
\text{Distance} = \sqrt{(4 - (-1))^2 + (2 - 4)^2} = \sqrt{(5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}
\]

The radius \(r\) is half of this distance:

\[
r = \frac{\sqrt{29}}{2}
\]

### Step 3: Write the standard equation of the circle
The standard equation of a circle with center \((h, k)\) and radius \(r\) is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Substituting the values \(h = \frac{3}{2}\), \(k = 3\), and \(r = \frac{\sqrt{29}}{2}\):

\[
\left(x - \frac{3}{2}\right)^2 + (y - 3)^2 = \left(\frac{\sqrt{29}}{2}\right)^2
\]

Simplifying the right side:

\[
\left(x - \frac{3}{2}\right)^2 + (y - 3)^2 = \frac{29}{4}
\]

Finally, multiply through by 4 to clear the fraction:

\[
4\left(x - \frac{3}{2}\right)^2 + 4(y - 3)^2 = 29
\]

Expanding and rearranging can give us the final standard form equation, but this form is generally acceptable.

### Final Answer:
The standard equation of the circle is:

\[
\left(x - \frac{3}{2}\right)^2 + (y - 3)^2 = \frac{29}{4}
\]

Would you like more details on any part of the solution? 

Here are some related questions for further exploration:
1. How would the equation change if the center were moved to the origin?
2. Can you convert this equation to the general form of a circle equation?
3. What are the coordinates of the other points on the circle?
4. How do you find the equation of a circle given the radius and center?
5. What is the relationship between the distance formula and the equation of a circle?
6. How do you derive the standard equation of a circle from the definition of a circle?
7. Can you find the equation of a circle with a center at (0, 0) and passing through (4, 2)?
8. How does the equation of a circle change if you rotate it about its center?

**Tip:** When solving geometry problems, always check if the given points can simplify the problem by aligning with the coordinate axes.

Learn how to find the standard equation of a circle with a diameter given endpoints A(-1, 4) and B(4, 2). This problem involves using the midpoint formula, distance formula, and standard form of circle equation. Suitable for Grades 9-12 students studying coordinate geometry.

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