Finding the Equation of a Circle Given Specific Points

The study of circles in geometry and algebra is fundamental in many fields. This article will guide you through the process of finding the equation of a circle given specific points. Specifically, we will find the equation of a circle passing through the points (3, 12), (3, 2), and (20, -5).

Introduction to the Problem

We are given three points that lie on a circle: (3, 12), (3, 2), and (20, -5). Our goal is to determine the equation of this circle. The general form of a circle's equation is:

[x^2 y^2 - 2hx - 2ky c 0]

where (h, k) is the center of the circle, and c is a constant. Alternatively, we can use the standard form:

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

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

Step-by-Step Solution

We will follow these steps to derive the equation of the circle:

Step 1: Substitute the Points

Substituting the given points into the equation of the circle, we get three equations:

For (3, 12): [3^2 12^2 - 3D - 12E F 0 implies 9 144 - 3D - 12E F 0 implies -3D - 12E F -153] For (3, 2): [3^2 2^2 - 3D - 2E F 0 implies 9 4 - 3D - 2E F 0 implies -3D - 2E F -13] For (20, -5): [20^2 (-5)^2 - 20D 5E F 0 implies 400 25 - 20D 5E F 0 implies -20D 5E F -425]

Step 2: Set Up the System of Equations

We now have the following system of equations:

-3D - 12E F -153 -3D - 2E F -13 -20D 5E F -425

Step 3: Solve the System

To solve for D, E, and F, we first subtract the second equation from the first:

[-3D - 12E F - (-3D - 2E F) -153 - (-13)]

This simplifies to:

[-10E -140 implies E 14]

Substituting E 14 into the second equation:

[-3D - 2(14) F -13 implies -3D - 28 F -13 implies -3D F 15 implies F 15 3D]

Substituting E 14 into the third equation:

[-20D 5(14) F -425 implies -20D 70 F -425 implies -20D F -495 implies F -495 20D]

Setting the two expressions for F equal:

[15 3D -495 20D implies 20D - 3D -495 - 15 implies 17D -510 implies D -30]

Substituting D -30 back into the expression for F:

[F 15 3(-30) 15 - 90 -75]

Thus, we have:

[D -30, E 14, F -75]

Step 4: Final Equation

The equation of the circle is:

[x^2 y^2 - 3 - 14y - 75 0]

This can be rewritten in the standard form:

[(x - 15)^2 (y - 7)^2 256]

The center of the circle is at (15, 7), and the radius is 16.

We hope this detailed solution helps you understand the process of finding the equation of a circle given specific points.

Conclusion

Mastering the technique of finding the equation of a circle is crucial for solving geometric and algebraic problems. By following these steps, you can confidently find the equation of a circle given any three points on its circumference. Remember to practice with different sets of points to solidify your understanding.