Write the standard form of the equation of the circle with endpoints of a diameter at the points

left parenthesis 1 comma 2 right parenthesis(1,2)

and left parenthesis negative 3 comma 5 right parenthesis .

To find the standard form of the equation of a circle, we need to know its center and radius. The standard form of the equation is:

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

where \((h, k)\) is the center and \(r\) is the radius.

### Step 1: Find the center of the circle
The center is the midpoint of the diameter. The midpoint \((h, k)\) of the points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:

\[
h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2}
\]

Substituting the given points \((1, 2)\) and \((-3, 5)\):

\[
h = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1
\]
\[
k = \frac{2 + 5}{2} = \frac{7}{2} = 3.5
\]

So, the center is \((-1, 3.5)\).

### Step 2: Find the radius
The radius is half the length of the diameter. The length of the diameter can be found using the distance formula:

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

Substituting the points \((1, 2)\) and \((-3, 5)\):

\[
d = \sqrt{((-3) - 1)^2 + (5 - 2)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

The radius \(r\) is half of the diameter:

\[
r = \frac{5}{2} = 2.5
\]

### Step 3: Write the equation in standard form
Now, we can substitute the center \((-1, 3.5)\) and the radius \(2.5\) into the standard form:

\[
(x + 1)^2 + (y - 3.5)^2 = (2.5)^2
\]

Simplifying the radius squared:

\[
(x + 1)^2 + (y - 3.5)^2 = 6.25
\]

Thus, the equation of the circle is:

\[
(x + 1)^2 + (y - 3.5)^2 = 6.25
\]

Would you like any further details or have any questions?

Here are 5 related questions:
1. How do you find the equation of a circle given its center and a point on the circle?
2. How do you convert the standard form of a circle equation to general form?
3. How can you find the length of a diameter if you only know the radius?
4. What is the relationship between a circle's diameter and circumference?
5. How do you find the center and radius of a circle from its general equation?

**Tip:** Always double-check calculations when using the distance formula, as small errors can lead to incorrect results.

Write the standard form of the equation of the circle with endpoints of a diameter at the points (1, 2) and (-3, 5).

Learn how to write the standard form of a circle's equation given the endpoints of its diameter at (1, 2) and (-3, 5). This solution covers key geometry concepts, including the midpoint formula, distance formula, and the equation of a circle. Suitable for students in Grades 9-11.

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