find the equation of the circle with A(2,-3) and B(-3,5) as end points of diamter

To find the equation of the circle with points \(A(2, -3)\) and \(B(-3, 5)\) as the endpoints of the diameter, follow these steps:

### Step 1: Find the center of the circle
The center of the circle is the midpoint of the diameter, which is the midpoint of points \(A\) and \(B\).

The midpoint \((h, k)\) of two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
\left(h, k\right) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

For points \(A(2, -3)\) and \(B(-3, 5)\):

\[
\left(h, k\right) = \left(\frac{2 + (-3)}{2}, \frac{-3 + 5}{2}\right)
\]

\[
\left(h, k\right) = \left(\frac{-1}{2}, \frac{2}{2}\right)
\]

\[
\left(h, k\right) = \left(-\frac{1}{2}, 1\right)
\]

So, the center of the circle is \(\left(-\frac{1}{2}, 1\right)\).

### Step 2: Find the radius of the circle
The radius \(r\) is half the length of the diameter. The length of the diameter is the distance between points \(A\) and \(B\), which can be found using the distance formula:

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

Substituting the coordinates of \(A(2, -3)\) and \(B(-3, 5)\):

\[
\text{Diameter} = \sqrt{(-3 - 2)^2 + (5 - (-3))^2}
\]

\[
\text{Diameter} = \sqrt{(-5)^2 + (8)^2} = \sqrt{25 + 64} = \sqrt{89}
\]

Thus, the radius \(r\) is:

\[
r = \frac{\sqrt{89}}{2}
\]

### Step 3: Write the equation of the circle
The general equation of the circle is:

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

Substituting the values of \(h = -\frac{1}{2}\), \(k = 1\), and \(r^2 = \left(\frac{\sqrt{89}}{2}\right)^2 = \frac{89}{4}\):

\[
\left(x + \frac{1}{2}\right)^2 + \left(y - 1\right)^2 = \frac{89}{4}
\]

To eliminate the fraction, multiply through by 4:

\[
4\left(x + \frac{1}{2}\right)^2 + 4\left(y - 1\right)^2 = 89
\]

Simplify further:

\[
\left(2x + 1\right)^2 + \left(2y - 2\right)^2 = 89
\]

This is the equation of the circle with \(A(2, -3)\) and \(B(-3, 5)\) as the endpoints of the diameter.

Would you like more details or have any other questions? Here are some related questions:

1. How to convert the equation of a circle to general form.
2. How to determine the length of a radius given the center and a point on the circle.
3. Applications of the midpoint formula in geometry.
4. Finding the equation of a circle given different forms of information.
5. Understanding the properties of a circle when the diameter is known.

**Tip:** When the endpoints of the diameter are given, always remember that the midpoint gives the center, and the distance between the endpoints provides the diameter, which can easily be halved to find the radius.

Learn how to find the equation of a circle using the endpoints of the diameter A(2,-3) and B(-3,5). Step-by-step guide includes finding the center using the midpoint formula, calculating the radius using the distance formula, and deriving the circle's equation. Suitable for high school geometry students.

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