Problem 95 - Algebra and Trigonometry

Question

Circle, Point, or Empty Set? Complete the squares in the general equation \(x^{2}+a x+y^{2}+b y+c=0,\) and simplify the result as much as possible. Under what conditions on the coefficients \(a, b,\) and \(c\) does this equation represent a circle? A single point? The empty set? In the case in which the equation does represent a circle, find its center and radius.

Step-by-Step Solution

Answer

Circle: if \(\frac{a^2}{4} + \frac{b^2}{4} > c\); Point: if \(\frac{a^2}{4} + \frac{b^2}{4} = c\); None: if \(\frac{a^2}{4} + \frac{b^2}{4} < c\).

Step 1: Identify the equation

We start with the given equation: \[x^2 + a x + y^2 + b y + c = 0\] Our aim is to complete the square for both the \(x\) and \(y\) terms.

Step 2: Completing the square for x

To complete the square for the \(x\) terms, take the coefficient of \(x\) which is \(a\), halve it, and then square it. This gives \((\frac{a}{2})^2\). Add and subtract this term:\[x^2 + a x = (x + \frac{a}{2})^2 - (\frac{a}{2})^2\]

Step 3: Completing the square for y

Next, complete the square for the \(y\) terms by taking the coefficient of \(y\) which is \(b\), halve it, and square it. This gives \((\frac{b}{2})^2\). Add and subtract this term:\[y^2 + b y = (y + \frac{b}{2})^2 - (\frac{b}{2})^2\]

Step 4: Substitute and simplify

Substitute the completed squares back into the equation:\[(x + \frac{a}{2})^2 - (\frac{a}{2})^2 + (y + \frac{b}{2})^2 - (\frac{b}{2})^2 + c = 0\]Simplify:\[(x + \frac{a}{2})^2 + (y + \frac{b}{2})^2 = \frac{a^2}{4} + \frac{b^2}{4} - c\]

Step 5: Determine conditions for different cases

This equation represents a **circle** if the right side is positive, a **single point** if it is zero, and the **empty set** if it is negative:- **Circle**: \(\frac{a^2}{4} + \frac{b^2}{4} - c > 0\)- **Point**: \(\frac{a^2}{4} + \frac{b^2}{4} - c = 0\)- **Empty Set**: \(\frac{a^2}{4} + \frac{b^2}{4} - c < 0\)

Step 6: Find center and radius of the circle

If the equation represents a circle, the center is \((-\frac{a}{2}, -\frac{b}{2})\) and the radius is \(\sqrt{\frac{a^2}{4} + \frac{b^2}{4} - c}\).

Key Concepts

Circle Equation

The circle equation is a fundamental part of geometry and algebra that helps us understand circular shapes analytically. In the context of this exercise, we start with a general quadratic equation:

• \(x^2 + ax + y^2 + by + c = 0\)

We aim to express this equation in a way that identifies it as a circle. To achieve this, we use the method of completing the square, which translates the equation into the standard form of a circle's equation. The resulting arrangement allows us to clearly see where the center of the circle is and how large it can be.
In the process, we rearrange and group the terms related to \(x\) and \(y\) to set them apart for completing the square method. This transformation is crucial as it lays the groundwork for figuring out the circle's characteristics and positioning in the coordinate plane.

Geometric Conditions

Understanding the geometric conditions for different mathematical objects is essential when solving equations like this one. By analyzing the equation

• \((x + \frac{a}{2})^2 + (y + \frac{b}{2})^2 = \frac{a^2}{4} + \frac{b^2}{4} - c\)

we can determine three scenarios:
1. **Circle**: If the expression on the right side \(\frac{a^2}{4} + \frac{b^2}{4} - c\) is greater than zero, the equation represents a circle.2. **Point**: When the right-hand side is exactly zero, it suggests a degenerate circle, which is actually a single point.3. **Empty Set**: If the expression is less than zero, it indicates an impossible geometric condition, thus representing an empty set with no solutions.
These conditions help us interpret the physical meaning of the equation, turning an abstract algebraic expression into tangible geometric possibilities.

Center and Radius of a Circle

Once the equation is identified as representing a circle, finding the center and radius becomes the next step. The form of the transformed equation,

• \((x + \frac{a}{2})^2 + (y + \frac{b}{2})^2 = R^2\)

indicates that the circle's center is located at the coordinates:

• \((-\frac{a}{2}, -\frac{b}{2})\)

The expression \(R^2\) in the equation represents the square of the circle's radius. Therefore, the radius \(R\) is

• \(\sqrt{\frac{a^2}{4} + \frac{b^2}{4} - c}\)

This insight into the circle's geometry allows us to visualize and define its size and location, making these algebraic reconstructions an integral part of understanding geometric shapes.

Algebraic Manipulation

Algebraic manipulation, particularly through completing the square, is how we transition from a general quadratic equation to one that reveals underlying geometric features. The process involves,

• Identifying quadratic terms and their linear components.
• Halving the linear coefficients \(a\) and \(b\), squaring them, and adjusting the equation by adding and subtracting these squares.

Through precise algebraic steps, as shown in the solution, we redefine the equation to a form that highlights the structure of a circle:

• \((x + \frac{a}{2})^2 + (y + \frac{b}{2})^2 = \frac{a^2}{4} + \frac{b^2}{4} - c\)

This manipulation simplifies the interpretation of the equation, effectively "drawing" the circle on the coordinate plane via pure algebra, turning numbers and variables into spatially meaningful information.