Introduction to Circle Equations

In this article, we will explore how to find the standard form of the equation of a circle when given the endpoints of a diameter. Understanding this concept is crucial in geometry and can be applied in various real-world scenarios, such as in engineering, architecture, and design.

Step-by-Step Guide to Finding the Circle Equation

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

Step 1: Determine the Center of the Circle

The center of the circle is the midpoint of the diameter. The midpoint formula is given by:

(frac{x_1   x_2}{2}, frac{y_1   y_2}{2})

Let's use the endpoints (1, 4) and (-3, 2) to find the center:

Calculation:

[h frac{1 (-3)}{2} frac{-2}{2} -1][k frac{4 2}{2} frac{6}{2} 3]

The center of the circle is at (-1, 3).

Step 2: Determine the Radius of the Circle

The radius is half the length of the diameter. The distance formula is used to find the length of the diameter:

sqrt{(x_2 - x_1)^2   (y_2 - y_1)^2}

Substituting the coordinates (1, 4) and (-3, 2):

[d sqrt{(-3 - 1)^2 (2 - 4)^2} sqrt{(-4)^2 (-2)^2} sqrt{16 4} sqrt{20} 2sqrt{5}]

The radius ( r ) is half of this distance:

[r frac{d}{2} frac{2sqrt{5}}{2} sqrt{5}]

So, the radius is (sqrt{5}).

Step 3: Write the Standard Form of the Circle Equation

The standard form of the equation of a circle is:

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

Substituting ( h -1 ), ( k 3 ), and ( r sqrt{5} ):

[(x - (-1))^2 (y - 3)^2 (sqrt{5})^2]

This simplifies to:

[(x 1)^2 (y - 3)^2 5]

The standard form of the equation of the circle is:

[boxed{(x 1)^2 (y - 3)^2 5}]

Example Walkthrough

Let's walk through finding the equation of a circle given endpoints (-14, 1) and (-32, 2).

Step 1: Find the Center

Using the midpoint formula:

[h frac{-14 (-32)}{2} frac{-46}{2} -23][k frac{1 2}{2} frac{3}{2} 1.5]

The center is at (-23, 1.5).

Step 2: Find the Radius

Using the distance formula:

[d sqrt{(-32 - (-14))^2 (2 - 1)^2} sqrt{(-18)^2 1^2} sqrt{324 1} sqrt{325} 5sqrt{13}]

The radius is:

[r frac{d}{2} frac{5sqrt{13}}{2}]

The radius is (frac{5sqrt{13}}{2}).

Step 3: Write the Equation

The standard form of the equation is:

[(x - (-23))^2 (y - 1.5)^2 left(frac{5sqrt{13}}{2}right)^2]

This simplifies to:

[(x 23)^2 (y - 1.5)^2 frac{25 times 13}{4} frac{325}{4}]

The standard form of the equation of the circle is:

[boxed{(x 23)^2 (y - 1.5)^2 frac{325}{4}}]

Conclusion

By understanding these steps and practicing with different endpoints, you can effectively find the standard form of the equation of a circle. This knowledge is not only theoretical but also has practical applications in several fields such as engineering, design, and architecture.

Further Reading and Practice

Explore additional resources and practice problems on the standard form of circle equations to solidify your understanding. You can find relevant exercises and tutorials in textbooks, online math forums, and educational websites.