Introduction to Finding the Equation of a Circle Through Given Points

In this article, we will explore the process of finding the equation of a circle that passes through three given key points, namely the origin and the points (2,0) and (3,-1). The derivation will be done using both the general form and the standard form of the circle equation, thereby providing a comprehensive understanding of the topic.

Equation of a Circle in General Form

The general form of a circle's equation is given by:

x ^2 y ^2 Dx Ey F 0

1. Deriving the Equation Passing Through the Origin

To find the circle equation that passes through the origin (0,0), we substitute these coordinates into the general form:

0 ^2 0 ^2 D ? 0 E ? 0 F 0

This simplifies to:`F` 0.

2. Deriving the Equation Using Additional Points

We substitute the additional points (2,0) and (3,-1) into the equation to find the values of D and E.

For the point (2,0):

2 ^2 0 ^2 D ? 2 E ? 0 0

This simplifies to: 4 2D 0, hence `D` -2.

For the point (3,-1):

3 ^2 -1 ^2 D ? 3 - E 0

This simplifies to: 9 1 - 3D - E 0. By substituting `D` -2, we get: 10 - 3(-2) - E 0, leading to: 10 6 - E 0, hence `E` 16.

3. Rearranging into Standard Form

The equation of the circle can be rearranged into the standard form:

x ^2 - 2x y ^2 - 4y 0

Completing the square for `x` and `y`:

x -1 ^2 x ^2 - 2x 1

y -2 ^2 y ^2 - 4y 4

Substitute back into the equation:

x -1 ^2 y -2 ^2 - 5 0

This simplifies to:

x -1 ^2 y -2 ^2 5

Conclusion

The equation of the circle, centered at (1,-2) with a radius of sqrt{5}, has been derived using the general and standard forms of the circle equation. This provides a comprehensive understanding of the circle properties and their application in solving for unknowns.