Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Use properties of determinants to show that the following is an equation of a circle through three noncollinear points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\), and \(\left(x_{3}, y_{3}\right)\) : $$ \left|\begin{array}{llll} x^{2}+y^{2} & x & y & 1 \\ x_{1}^{2}+y_{1}^{2} & x_{1} & y_{1} & 1 \\ x_{2}^{2}+y_{2}^{2} & x_{2} & y_{2} & 1 \\ x_{3}^{2}+y_{3}^{2} & x_{3} & y_{3} & 1 \end{array}\right|=0 $$

Step-by-Step Solution

Answer

The determinant represents a circle passing through the three points, due to the zero condition forming a circle's equation based on these coordinates.

Step 1: Understand the determinant structure

The determinant you are given involves the coordinates of three points and a generic point \( (x, y) \). Each row except the first contains a specific point. The function of the determinant is for it to be zero if \( (x, y) \) lies on the circle passing through the three given points.

Step 2: Recall the equation of a circle

A circle's equation in standard form is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.

Step 3: Expand the determinant by minors

To verify if the determinant represents a circle, expand it along the first row. This results in a combination of minors involving the x, y, and constant terms against the respective coordinates of the points.

Step 4: Simplify and interpret the minors

The minor determinants will form linear equations when solved for each \( x^2 + y^2 \), x, and y term. The combination of these linear expressions reflects the coefficients in the general circle equation.

Step 5: Conclusion from determinant properties

When the determinant is zero, it indicates that \( (x, y) \), in terms of \( x^2 + y^2 \), x, and y, the expressions equal zero if and only if \( (x, y) \) lies on the circle defined by these constraints. Thus, the given determinant equation represents a circle through these three points.

Key Concepts

Circle Equation

A circle is a set of all points in a plane that are at a fixed distance from a given point, known as the center. The equation for a circle in its standard form is \[(x - h)^2 + (y - k)^2 = r^2\] where

• \((h, k)\) is the center of the circle
• \(r\) is the radius of the circle

However, determining the equation for a circle through three noncollinear points requires a different approach. The use of determinants is a clever mathematical method to verify if points are collinear and applies to problems such as determining circles. In this exercise, a specific determinant is set to zero to derive the circle's equation, highlighting the points' role in defining geometric properties.

Noncollinear Points

The term "noncollinear" refers to points that do not all lie on the same straight line. Understanding whether points are noncollinear is essential when solving geometric problems involving circles. If three points are collinear, they lie on the same line, and a circle cannot pass through all three simultaneously. Hence, it's fundamental to ensure that the points are noncollinear so they can form a unique circle.
To check this, a determinant structure set to zero confirms the circle's equation through the given points. The determinant demonstrates the geometric condition that makes these points form a circle, rather than a single line or any other shape. This property ensures that the circle truly exists and is well-defined when passing through these specific points.

Geometric Properties

The geometric properties related to circles are essential in understanding circle equations through points. The relation of points within a plane regarding a circle involves their distances and central symmetry around the center. In the context of the determinant problem, we harness these geometric properties to effectively show that the expression is indeed a circle.
Using the determinant, we establish the conditions under which a set of points will define a unique circular path. The notion of distance from the circle center and the symmetry in points are crucial for identifying the circle properly. These properties ensure consistent characteristics of a circle, such as equidistant points from the center, which needs to be satisfied.

Determinant Expansion

Determinant expansion is a method used to decompose a determinant into its smaller components, making it easier to solve or simplify. The provided exercise utilizes the expansion of a 4x4 determinant to establish the circle's equation. When expanding a determinant along the first row, minor determinants are introduced. Each minor is a smaller determinant obtained by removing one row and one column from the original matrix.

• The expansion shows that the result reflects linear relationships between the coordinates (x, y) and those of the given points.
• These relationships correspond to the components of the circle's equation centered around the three noncollinear points.

By simplifying these minors and equating them to zero, we directly link the determinant's properties to the geometric constraints of a circle. This method proves to be insightful in connecting abstract algebraic concepts to geometric visualization of a circle.