Problem 65 - A Graphical Approach to College Algebra

Question

Given three noncollinear points, there is one and only one circle that passes through them. Knowing that the equation of a circle may be written in the form. $$x^{2}+y^{2}+a x+b y+c=0$$ $$\begin{aligned} &\text {find the equation of the circle passing through the points}\\\ &\text { shown or specified.} \end{aligned}$$ (GRAPH CAN'T COPY)

Step-by-Step Solution

Answer

Use given points to derive circle's equation by solving linear equations.

Step 1: Identify the Three Points

Let's assume the three given points are \(P_1(x_1, y_1)\), \(P_2(x_2, y_2)\), and \(P_3(x_3, y_3)\) that lie on the circle. These represent the x and y coordinates of each point through which the circle passes.

Step 2: Substitute Points into Circle Equation

We substitute each point into the equation of the circle \(x^2 + y^2 + ax + by + c = 0\) to create three separate equations. For point \((x_1, y_1)\): \(x_1^2 + y_1^2 + ax_1 + by_1 + c = 0\) For point \((x_2, y_2)\): \(x_2^2 + y_2^2 + ax_2 + by_2 + c = 0\) For point \((x_3, y_3)\): \(x_3^2 + y_3^2 + ax_3 + by_3 + c = 0\).

Step 3: Set Up the System of Linear Equations

Write out the system of equations derived from the substitution: 1) \(ax_1 + by_1 + c = -(x_1^2 + y_1^2)\) 2) \(ax_2 + by_2 + c = -(x_2^2 + y_2^2)\) 3) \(ax_3 + by_3 + c = -(x_3^2 + y_3^2)\).

Step 4: Solve the System of Equations

Using linear algebra techniques (such as substitution, elimination, or matrix methods), solve this system for the unknowns \(a\), \(b\), and \(c\).

Step 5: Write the Equation of the Circle

Once you have the values for \(a\), \(b\), and \(c\), substitute them back into the circle equation to get the final equation: \(x^2 + y^2 + ax + by + c = 0\).

Key Concepts

Noncollinear Points

Noncollinear points are essential in geometry to ensure uniqueness when defining certain shapes like circles. Noncollinear points are three or more points that do not all lie on a single straight line. This condition is important because if you consider three collinear points, they would all lie on a line and no unique circle could pass through all three simultaneously. A circle requires curvature, which would inherently contradict the linear alignment of collinear points.

When tasked with finding a circle that passes through three points, confirming they are noncollinear ensures that a unique circle can indeed exist. Geometrically, any set of three noncollinear points can define a unique circle. This is because there is exactly one circle that can meet the curvature and span through these points, giving a definitive geometric solution.

Identifying these noncollinear points is the first step in constructing the equation of a circle and ensures the validity of the process that follows.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. In the context of finding the equation of a circle through three noncollinear points, substituting the coordinates into the general circle equation results in multiple linear equations.

Here, substituting each of the three given noncollinear points into the circle equation results in three linear equations. Each equation arises by substituting the x and y coordinates of each point into the generic circle equation form:

• Equation for Point 1: \(ax_1 + by_1 + c = -(x_1^2 + y_1^2)\)
• Equation for Point 2: \(ax_2 + by_2 + c = -(x_2^2 + y_2^2)\)
• Equation for Point 3: \(ax_3 + by_3 + c = -(x_3^2 + y_3^2)\)

This results in a system of equations with three unknowns: \(a\), \(b\), and \(c\).

The consistency and non-redundancy of these equations ensure that, when solved, they provide a unique set of values for the coefficients \(a\), \(b\), and \(c\), essential for correctly defining the circle through the given points.

Linear Algebra Techniques

Linear algebra techniques provide powerful methods to solve the system of linear equations derived from the circle equation. Common methods include substitution, elimination, and matrix operations.

• Substitution: This involves solving one equation for a variable and substituting that expression into the other equations. It simplifies the system by progressively reducing the number of involved variables.
• Elimination: By adding or subtracting equations, variables are systematically eliminated, simplifying the system down to fewer variables until isolated equations for each variable remain.
• Matrix Methods: Using matrices, the system can be expressed as \(A \vec{x} = \vec{b}\), where \(A\) is the coefficient matrix, \(\vec{x}\) is a vector of unknowns \([a, b, c]\), and \(\vec{b}\) is the constant vector. Matrix techniques such as Gaussian elimination or using inverse matrices can efficiently solve these systems, particularly beneficial for larger systems.

By employing these techniques, values for \(a\), \(b\), and \(c\) are obtained, which complete the circle's equation. Mastery of these linear algebra techniques significantly enhances the ability to solve similar geometric and mathematical problems.