Problem 49 - Precalculus

Question

Sketch (if possible) the graph of the degenerate conic. $$x^{2}-2 x y+y^{2}=0$$

Step-by-Step Solution

Answer

The given equation \(x^{2}-2 x y+y^{2}=0\) happens to be a degenerate conic in the form of a straight line represented by the equation \(y = x\). This line passes through the origin in the 2 dimension plane.

Step 1: Rewrite the Equation

We initially identify the given equation as quadratic. The equation can be written in a different form \( (x-y)^2 =0 \) by grouping the terms.

Step 2: Solve for (x-y)

Upon squaring both sides of the equation, we find \( x-y = 0 \), which implies \( y = x \).

Step 3: Identify the type of Conic and Plot

The equation \( y = x \) is a linear equation presented as a straight line passing through the origin. This means our quadratic equation represents a degenerate conic section in the form of a line. The degenerate conic is plotted in the 2D plane.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2, generally written in the form \( ax^2 + bx + c = 0 \).

• It includes terms where the highest exponent of the variable \( x \) is 2.
• Quadratic equations can describe a variety of curves, including parabolas, circles, ellipses, and hyperbolas.
• These equations are fundamental in algebra and geometry.

When dealing with conic sections, quadratic equations often classify different types of curves based on their coefficients. For instance, the given quadratic equation \( x^2 - 2xy + y^2 = 0 \) is somewhat different because it represents a degenerate conic. By manipulating the terms of the equation into \( (x-y)^2 = 0 \), we can simplify it to a form where it becomes evident that there’s no second-degree curve, but rather a linear one. This linear form indicates the equation's degenerate nature, indicating a simpler, reduced form such as a line or a point.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. There are four main types:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

How the plane intersects the cone determines the type of conic. When a quadratic equation like \( x^2 - 2xy + y^2 = 0 \) transforms into a simpler form—because specific terms cancel each other out—it becomes a degenerate conic. This is not one of the typical conic sections such as a parabola or circle but rather something simpler. In this case, the equation reduces to \( y = x \), showing that our degenerate conic is actually a line. This transformation shows the unique nature of conics, where the intersection's angle and position often dictate the outcome shape.

Graph of a Line

The graph of a line is one of the simplest forms you can encounter in 2D geometry. It is represented by a linear equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• In our case, the equation \( y = x \) represents a line with a slope of 1 and a y-intercept of 0.
• Such a line passes through the origin \((0,0)\) and forms a 45-degree angle with both the x-axis and y-axis.

To sketch this, simply find a couple of points that satisfy the equation. The simplest are where \( x = y \). Drawing through these points provides a visual confirmation of the equation's representation as a line. Understanding lines is crucial, as they serve as a foundation for more complex geometric figures and problems.

2D Geometry

Two-dimensional or 2D geometry involves shapes, sizes, and the relative positions of figures in a plane. It covers concepts like points, lines, and polygons:

• A point signifies a location in space without dimensions.
• A line is a continuous extent of points.
• Conic sections fit within this scope as they describe curves in the plane.

The exercise involves sketching a degenerate conic, specifically \( y = x \), which takes place entirely in 2D. To graph this line in 2D geometry, you only need an x and y-axis. Start plotting from the origin, marking equal distance in both horizontal and vertical directions. This simplicity exemplifies the broader scope of 2D geometry, where even complex equations can reduce down to simple, interpretable graphics. Mastery of 2D geometry is often a stepping stone to understanding more advanced mathematical concepts.