Find the standard form for the equation of a circle (x-h)^2 +(y-k)^2 = r^2 with a diameter that has endpoints (-5,10) and (7,4)

To find the standard form of the equation of a circle, we need to determine the center \((h, k)\) and the radius \(r\). The center is the midpoint of the diameter, and the radius is half the length of the diameter.

### Step 1: Find the center \((h, k)\)

The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Using the given endpoints \((-5, 10)\) and \((7, 4)\):

\[
h = \frac{-5 + 7}{2} = \frac{2}{2} = 1
\]
\[
k = \frac{10 + 4}{2} = \frac{14}{2} = 7
\]

So, the center is \((1, 7)\).

### Step 2: Find the radius \(r\)

The radius is half the length of the diameter. First, we calculate the length of the diameter using the distance formula:

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

Substituting the coordinates:

\[
d = \sqrt{(7 - (-5))^2 + (4 - 10)^2} = \sqrt{(7 + 5)^2 + (-6)^2} = \sqrt{12^2 + (-6)^2} = \sqrt{144 + 36} = \sqrt{180}
\]

Thus, the diameter is \(\sqrt{180}\), so the radius \(r\) is:

\[
r = \frac{\sqrt{180}}{2} = \frac{\sqrt{36 \cdot 5}}{2} = \frac{6\sqrt{5}}{2} = 3\sqrt{5}
\]

### Step 3: Write the equation in standard form

Now that we know the center \((h, k) = (1, 7)\) and the radius \(r = 3\sqrt{5}\), we can write the equation of the circle in standard form:

\[
(x - 1)^2 + (y - 7)^2 = (3\sqrt{5})^2
\]

Simplifying the right side:

\[
(x - 1)^2 + (y - 7)^2 = 45
\]

### Final Answer:

The standard form of the equation of the circle is:

\[
(x - 1)^2 + (y - 7)^2 = 45
\]

Would you like further details or have any questions?

---

Here are 5 related questions to explore:

1. How would you find the equation if only the center and a point on the circle were given?
2. What is the general form of the equation of a circle, and how does it differ from the standard form?
3. How do you find the area and circumference of a circle using its radius?
4. How would the equation change if the circle were translated or reflected?
5. Can you derive the equation of a circle given its radius and tangent line?

**Tip:** When dealing with circles, always remember that the radius is the distance from the center to any point on the circumference.

Learn how to find the standard form equation of a circle with endpoints (-5, 10) and (7, 4) as the diameter. This solution covers essential geometry concepts, including the midpoint and distance formulas, and demonstrates the step-by-step process to derive the circle's equation. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation