Determine the general equation of: endpoints of the diameter are (0,0) and (3,2)

To determine the general equation of a circle given the endpoints of a diameter, you can follow these steps:

### 1. Find the center of the circle:
The center of the circle is the midpoint of the diameter. The formula for the midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Substituting the given points \((0, 0)\) and \((3, 2)\):

\[
\text{Center} = \left(\frac{0 + 3}{2}, \frac{0 + 2}{2}\right) = \left(\frac{3}{2}, 1\right)
\]

### 2. Find the radius of the circle:
The radius is half the length of the diameter. The length of the diameter can be found using the distance formula between the two points:

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

Substituting the points:

\[
\text{Diameter length} = \sqrt{(3 - 0)^2 + (2 - 0)^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}
\]

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

\[
r = \frac{\sqrt{13}}{2}
\]

### 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 the center \(\left(\frac{3}{2}, 1\right)\) and the radius \(\frac{\sqrt{13}}{2}\):

\[
\left(x - \frac{3}{2}\right)^2 + (y - 1)^2 = \left(\frac{\sqrt{13}}{2}\right)^2
\]

Simplifying the equation:

\[
\left(x - \frac{3}{2}\right)^2 + (y - 1)^2 = \frac{13}{4}
\]

Multiplying through by 4 to eliminate the fraction:

\[
4\left(x - \frac{3}{2}\right)^2 + 4(y - 1)^2 = 13
\]

Expanding and simplifying:

\[
(x - \frac{3}{2})^2 = x^2 - 3x + \frac{9}{4}, \quad 4(x - \frac{3}{2})^2 = 4x^2 - 12x + 9
\]

\[
(y - 1)^2 = y^2 - 2y + 1, \quad 4(y - 1)^2 = 4y^2 - 8y + 4
\]

Adding both expressions:

\[
4x^2 - 12x + 9 + 4y^2 - 8y + 4 = 13
\]

Simplifying the final equation:

\[
4x^2 + 4y^2 - 12x - 8y + 13 = 13
\]

Subtracting 13 from both sides:

\[
4x^2 + 4y^2 - 12x - 8y = 0
\]

Or:

\[
x^2 + y^2 - 3x - 2y = 0
\]

This is the general equation of the circle.

Would you like further details or have any questions?

---

Here are 5 related questions:
1. How do you find the general form of a circle equation given the center and radius?
2. What is the distance formula and when is it used?
3. How do you convert the general form of a circle equation to the standard form?
4. How do you determine if a point lies on a circle?
5. How do you find the length of the radius if you have the endpoints of the diameter?

**Tip:** Always verify your final equation by substituting the original points to ensure they satisfy the equation.

Determine the general equation of a circle: endpoints of the diameter are (0,0) and (3,2)

Discover how to find the general equation of a circle using the endpoints of its diameter at (0,0) and (3,2). This problem covers key geometric concepts, including the midpoint formula, distance formula, and circle equation. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation