Problem 85 - 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 that the equation does represent a circle, find its center and radius.

Step-by-Step Solution

Answer

The equation represents a circle if \( c < \frac{a^2}{4} + \frac{b^2}{4} \), its center is \((-\frac{a}{2}, -\frac{b}{2})\), and radius is \( \sqrt{\frac{a^2}{4} + \frac{b^2}{4} - c} \). It is a point if \( c = \frac{a^2}{4} + \frac{b^2}{4} \), and empty if \( c > \frac{a^2}{4} + \frac{b^2}{4} \).

Step 1: Rewrite the General Equation

Start by rewriting the given general equation of the conic section: x^{2}+ax+y^{2}+by+c=0. Our goal is to put this equation into the standard form by completing 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 \(x^2 + ax\), compute the term needed:\(\left(\frac{a}{2}\right)^2 = \frac{a^2}{4}\). So, add and subtract \(\frac{a^2}{4}\) inside the equation.This gives us \( (x + \frac{a}{2})^2 - \frac{a^2}{4} \).

Step 3: Completing the Square for y

Similarly, complete the square for the \(y\) terms, by looking at \(y^2 + by\):\(\left(\frac{b}{2}\right)^2 = \frac{b^2}{4}\).Add and subtract \(\frac{b^2}{4}\) so it becomes \( (y + \frac{b}{2})^2 - \frac{b^2}{4} \).

Step 4: Substitute and Simplify

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

Step 5: Determine Conditions for a Circle

For the equation to represent a circle, the right side must be positive:\(\frac{a^2}{4} + \frac{b^2}{4} - c > 0\). This indicates the sum of the squares must exceed \(c\).

Step 6: Determine Conditions for a Point

The equation represents a single point when the right side of the equation is zero:\(\frac{a^2}{4} + \frac{b^2}{4} - c = 0\).This means \(c = \frac{a^2}{4} + \frac{b^2}{4}\).

Step 7: Determine Conditions for the Empty Set

The equation represents the empty set when the right side of the equation is negative:\(\frac{a^2}{4} + \frac{b^2}{4} - c < 0\).This means \(c > \frac{a^2}{4} + \frac{b^2}{4}\).

Step 8: Identify the Center and Radius of the Circle

If the equation is a circle, its center is at:\((-\frac{a}{2}, -\frac{b}{2})\).The radius \(r\) is given by \(r = \sqrt{\frac{a^2}{4} + \frac{b^2}{4} - c}\), as long as \(\frac{a^2}{4} + \frac{b^2}{4} - c > 0\).

Key Concepts

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is crucial for putting equations of conic sections into a recognizable form, which makes it easier to determine their properties such as centers and radii of circles.

To complete the square in an equation such as \(x^2 + ax + y^2 + by = 0\), we focus on the \(x\) and \(y\) terms separately. For the \(x\) terms, rewrite \(x^2 + ax\) as \((x + \frac{a}{2})^2 - \frac{a^2}{4}\). Similarly, for the \(y\) terms, express \(y^2 + by\) as \((y + \frac{b}{2})^2 - \frac{b^2}{4}\).

By adding and subtracting these squares within the equation, we transform the expression into a form where both \(x\) and \(y\) terms are perfect squares, helping to easily identify the circle's features.

Conditions for a Circle

The equation representing a circle, in its general form, must satisfy specific conditions. For the transformed equation \((x + \frac{a}{2})^2 + (y + \frac{b}{2})^2 = \frac{a^2}{4} + \frac{b^2}{4} - c\), the right side must be positive for it to describe a real circle. This is because

• A positive right side signifies that the equation has a radius that is a real number.
• The condition \(\frac{a^2}{4} + \frac{b^2}{4} - c > 0\) ensures the circle exists.

If this condition is not met, the equation may represent other geometric situations such as a single point or no real solution at all.

Center and Radius of a Circle

Once you've completed the square, the circle's equation takes the standard form \((x + \frac{a}{2})^2 + (y + \frac{b}{2})^2 = r^2\). Here, the circle's center and radius are neatly revealed.

• The center of the circle is found at the point \((\frac{-a}{2}, \frac{-b}{2})\).

The radius \(r\) can be calculated using the equation \(r = \sqrt{\frac{a^2}{4} + \frac{b^2}{4} - c}\). Remember, the radius is only valid and real if \(\frac{a^2}{4} + \frac{b^2}{4} - c > 0\). These components reflect the geometrical aspects of the circle derived from the equation.

General Equation of Conic Sections

The general equation for conic sections encompasses all shapes like circles, ellipses, parabolas, and hyperbolas, typically represented as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Circles are a special case within this framework.

When analyzing conics specifically for a circle using the general form \(x^2 + ax + y^2 + by + c = 0\), the equation simplifies after completing the square. Such simplification highlights the specific conditions under which the conic section is a circle.

Understanding the broader family of conic sections helps appreciate how circles fit into this classification, being defined by the condition of equal coefficients for both squared terms and no xy-cross product.