Problem 53 - Precalculus with Limits

Question

In Exercises 51-56, sketch (if possible) the graph of the degenerate conic. \(x^2-2xy+y^2=0\)

Step-by-Step Solution

Answer

The graph of the equation \(x^2-2xy+y^2=0\) is a straight line where every point has equal x and y coordinates, angled at 45 degrees and passing through the origin.

Step 1: Identify type of conic

We can simply determine the type of the conic by looking at the equation: \( x^2-2xy+y^2 = 0 \). By comparing it to the general form of a conic section \( Ax^2 + Bxy + Cy^2 = 0 \), if \( B^2 - 4AC = 0 \), then it is a parabola, if \( B^2 - 4AC < 0 \), then it is an ellipse or a circle, and if \( B^2 - 4AC > 0 \), then it is hyperbola. In our case, \( B = -2, A = C = 1 \) so \( B^2 - 4AC = 0 \), which indicates our equation is for degenerate conics.

Step 2: Rewrite the equation in a recognizable form

To determine if the degenerate conic is a pair of intersecting lines, two parallel lines, a single line, or a point, we need to rewrite the equation into a more recognizable form. Let's factor the left side of the equation: \( (x-y)^2 = 0 \)

Step 3: Sketch the graph

When a square equals to zero, it means the term inside the square must be zero. So \( x - y = 0 \), which is a straight line angled at 45 degrees and passing through the origin. Thus, the graph of this equation is simply a straight line where every point has equal x and y coordinates.

Key Concepts

Conic Sections Classification

Conic sections are the curves obtained by slicing a cone with a plane at different angles, resulting in a variety of shapes such as circles, ellipses, parabolas, and hyperbolas. Each of these shapes has a distinctive set of properties and equations that describe them.

To classify a conic section from its quadratic equation, like the one in our exercise, we analyze the coefficients and discriminate of the equation in the form of \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The discriminate \( B^2 - 4AC \) is crucial for classifying the conic:

• If \( B^2 - 4AC > 0 \), the conic is a hyperbola.
• If \( B^2 - 4AC = 0 \), the conic is a parabola, or it might be degenerate.
• If \( B^2 - 4AC < 0 \), the conic is an ellipse or a circle.

A degenerate conic occurs when the conic section seems to 'degenerate' into a simpler shape, which could be a point, a line, or two intersecting lines, depending on the specific equation given. In the case of the exercise, the given equation \(x^2-2xy+y^2=0\) indicates a degenerate conic since \( B^2 - 4AC = (-2)^2 - 4(1)(1) = 0 \).

Factoring Quadratic Equations

Factoring quadratic equations is a method used to rewrite the expressions in a product form that reveals more information about the equation's graphical representation and solutions. When dealing with second-degree polynomials, factoring can often simplify complex equations and reveal their roots.

For the quadratic equation \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, we look for two numbers that multiply to \( ac \) and add to \( b \). However, with a conic section equation like in our exercise, \( x^2 - 2xy + y^2 = 0 \), we can use the method of factoring by grouping or recognize it as a perfect square trinomial. The trinomial factors into \( (x - y)^2 = 0 \), which significantly simplifies the equation and reveals its nature – in this case, a degenerate conic.

Sketching Conic Section Graphs

The ability to sketch the graph of conic sections is a valuable skill. It provides a visual representation of the solutions to the equation, which is essential for understanding their behavior. To begin sketching, we need to simplify the equation as much as possible, ideally into a factored form, which can reveal intercepts and symmetry.

In our example, \( (x-y)^2 = 0 \) simplifies to \( x - y = 0 \), which is a linear equation describing a straight line where \( x \) equals \( y \). When we plot such an equation, we draw a line at a 45-degree angle to both the x-axis and y-axis, passing through the origin. This line indicates that for any point on it, the x and y coordinates are equal.

Remember, when dealing with degenerate conics like in this exercise, the graph may not be a curve but rather a single line or even just a point. Recognizing the factored form expedites the graphing process and augments comprehension.