Question

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) x^{2}+y^{2}-2 x+2 y+1=0

Short answer

The given equation represents a point at (1, -1).

Step-by-step solution

Group Terms

First, let's group the terms that involve the same variable. Split the quadratic terms and constant term: \(x^{2} - 2x + y^{2} + 2y = -1\).

Complete the Square for x

For the terms involving \(x\), complete the square. Take the coefficient of \(x\), which is \(-2\), divide it by 2 (getting \(-1\)), and square it (getting \(1\)). Add and subtract this square inside the equation: \((x^{2} - 2x + 1) + y^{2} + 2y = -1 + 1\).

Complete the Square for y

Similarly, complete the square for \(y\). The coefficient of \(y\) is \(2\). Divide it by 2 (getting \(1\)) and square it (getting \(1\)). Add and subtract this square: \((x^{2} - 2x + 1) + (y^{2} + 2y + 1) = 0\).

Rewrite as Perfect Squares

Now, rewrite the quadratic expressions as perfect squares: \((x - 1)^{2} + (y + 1)^{2} = 0\).

Identify the Conic Section

The equation \((x - 1)^{2} + (y + 1)^{2} = 0\) represents a point because both squared terms add to zero. The point is \((1, -1)\).

Key concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique is particularly useful in algebra to make solving and understanding quadratic equations easier. Here's how you can complete the square:1. Start with the quadratic expression you want to transform, such as \(x^2 + bx\). - Group related terms together and isolate the constant term, if necessary. 2. Take the coefficient of the linear term (in this example, it's the coefficient \(b\)), divide it by 2, and square the result. - For instance, if your expression is \(x^2 - 2x\), you take \(-2\), divide by 2 to get \(-1\), and then square it to obtain \(1\).3. Add and subtract this squared number from your expression. - This step ensures the algebraic balance of the equation and will help to express the equation as a perfect square.4. Rewrite the expression as a squared binomial. - Following our example, \(x^2 - 2x + 1\) becomes \((x-1)^2\).Completing the square simplifies equations and is also a vital process in coordinate geometry, especially when identifying types of conic sections like circles and ellipses.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2. They take the general form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants.Key points about quadratic equations:

• They can have two, one, or no real roots depending on the discriminant \(b^2 - 4ac\).
• The solutions to quadratic equations can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Graphically, they represent a parabola in the coordinate plane that opens upwards if \(a > 0\) and downwards if \(a < 0\).

Completing the square is one method used to solve quadratic equations. By converting them into a form \((x + p)^2 = q\), you can easily determine the roots of the equation.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves using algebraic methods to solve geometric problems based on a coordinate plane. This branch of geometry allows us to describe conic sections, such as circles, parabolas, ellipses, and hyperbolas, using algebraic equations.

• Each point on the coordinate plane is described by an ordered pair \((x, y)\).
• Geometric shapes are given specific equations, enabling graphical representation.
• Distance, midpoint, and slope are fundamental concepts that help analyze relative positions and inclinations of lines and shapes.

Conic sections can be described using quadratic equations. For instance, converting a general quadratic equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) into its standard form, often through completing the square, allows us to identify the type of conic section that is represented. Understanding the properties of these figures enhances problem-solving skills and the visualization of relationships in geometry.