Understanding the Equation of a Circle Passing Through Given Points

In this article, we will investigate the process of finding the equation of a circle that passes through specific points. This involves understanding the relationships between points, the diameter of the circle, and the equations derived from these points.

Introduction

Given three points, A(2, -1), B(-3, 0), and C(1, 4), we aim to find the equation of the circle that passes through these points. This will involve using principles from coordinate geometry and the properties of circles.

Method and Steps

We will use two different methods to find the equation of the circle:

Direct Method: Using the perpendicular bisector and family of circles. General Method: Using the general equation of a circle.

Direct Method

The direct method involves the following steps:

Find the midpoint of the diameter AB. Determine the slope of AB to find the slope of the perpendicular bisector. Use the family of circles passing through A and B to find the required circle.

1. The midpoint of AB is given by:

$$left(frac{2 -3}{2}, frac{-1 0}{2}right) left(-frac{1}{2}, -frac{1}{2}right)$$

2. The slope of AB is given by:

$$text{slope} frac{0 - (-1)}{-3 - 2} -frac{1}{5}$$

3. The slope of the perpendicular bisector is the negative reciprocal of (-frac{1}{5}), which is 5.

4. The equation of the chord AB is:

$$x 5y 3 0$$

5. The family of circles passing through A and B can be written as:

$$x^2 y^2 x y - 6 k(x 5y 3) 0$$

6. Substituting the coordinates of C(1, 4) into the family of circles, we get:

$$1^2 4^2 1 4 - 6 k(1 20 3) 0$$ $$25 25k 0$$ $$k -frac{5}{5} -1$$

7. Substituting (k -frac{2}{3}) into the family equation, we get the equation of the required circle:

$$x^2 y^2 x y - 6 - frac{2}{3}(x 5y 3) 0$$ $$3x^2 3y^2 - x - 7y - 24 0$$

General Method

The general method involves using the general equation of a circle (x^2 y^2 2gx 2fy c 0).

Using point A(2, -1), we get: $$4 1 4g - 2f c 0$$ $$4g - 2f c -5$$ Using point B(-3, 0), we get: $$9 - 6g c 0$$ $$-6g c -9$$ Using point C(1, 4), we get: $$1 16 2g 8f c 0$$ $$2g 8f c -17$$

Using these three equations:

$$4g - 2f c -5………1$$ $$-6g c -9………2$$ $$2g 8f c -17………3$$

From 2: (c 6g - 9), substitute in 1 and 3:

$$4g - 2f 6g - 9 -5$$ $$-6g - 9 2g 8f - 17 0$$

From 1 and 3:

$$18g 5c -37………4$$ $$8c -64$$ $$c -8$$

Using (c -8) in 2:

$$-6g - 8 -9$$ $$g -frac{1}{6}$$

Using (g -frac{1}{6}) and (c -8) in 1:

$$4(-frac{1}{6}) - 2f - 8 -5$$ $$-2f - frac{2}{3} - 8 -5$$ $$-2f - frac{28}{3} -5$$ $$2f -frac{13}{3}$$ $$f -frac{7}{6}$$

The equation of the circle is:

$$x^2 y^2 - frac{1}{3}x - frac{7}{3}y - 8 0$$ $$3x^2 3y^2 - x - 7y - 24 0$$

Conclusion

The process of finding the equation of a circle passing through given points involves a combination of geometric principles and algebraic manipulation. By using the perpendicular bisector of a chord and the general equation of a circle, we can determine the equation of the circle. This method is versatile and can be applied to various problems in coordinate geometry and circle theory.