Problem 56 - Precalculus with Limits

Question

In Exercises 51-56, sketch (if possible) the graph of the degenerate conic. \(x^2-10xy+y^2=0\)

Step-by-Step Solution

Answer

The given degenerate conic is a pair of intersecting lines. The lines have a slope of 1/5 and intersect at the origin, so sketch a line that passes through the origin with this slope.

Step 1: Identifying the Conic

The conic \(x^2-10xy+y^2=0\) is degenerate, it is a pair of intersecting lines. This can be determined by rewriting the equation in the standard form of a degenerate conic section, that is either a single point, straight line or pair of straight lines.

Step 2: Rewriting into Standard Form

Rewrite the given equation in the standard form, \(ax^2+bxy+cy^2=0\), and determine the value of \(b^2-4ac\). In this equation, \(a=c=1\), \(b=-10\). So, \(b^2-4ac=(-10)^2-4*1*1=100-4=96>0\). According to the conic theory, when \(b^2-4ac>0\), the equation of conic section represents a real and distinct pair of straight lines.

Step 3: Finding the Line Equations

To find the intersecting lines, we need to factorize the given equation and represent it in the form of two linear equations. Let's factorize: \(x^2-10xy+y^2=0\) => (x-5y)(x-5y)=0. Hence the two intersecting lines are x-5y=0 and x-5y=0.

Step 4: Sketching the Graph

Both our lines overlap each other, form a line that has a slope of 1/5 and passes through the origin. Because x and y are both equal to 0 in these equations, the lines intersect at the origin.

Key Concepts

Conic Sections

Conic sections are the curves obtained as the intersection of the surface of a cone with a plane. They are fundamental to the study of analytic geometry and have diverse applications in fields such as physics, engineering, and astronomy. The four basic types of conic sections are circles, ellipses, hyperbolas, and parabolas.

When special conditions are met, such as when a conic does not represent two distinct curves but rather degenerates into a simpler one, we get what are known as degenerate conics. These can be single points, straight lines, or a pair of intersecting lines, which occur when the conic equation does not define an area. In our exercise, the given conic is a degenerate conic as it represents a pair of intersecting lines.

Factoring Equations

Factoring is an algebraic process of expressing an equation as a product of its factors. For conic section equations, factoring can simplify the representation and help identify its type. In the case of the given degenerate conic, the equation was factored into the form \(x-5y)(x-5y)=0\), highlighting that it comprises two identical linear factors.

Understanding factoring is essential for simplifying complex expressions and solving equations efficiently. It's a pivotal skill for analyzing conic sections and is also frequently applied in calculus and other areas of mathematics.

Graphing Conics

Graphing conics involves plotting the curves or points described by their equations onto a coordinate plane. The strategy often requires converting the conic's equation to a recognizable standard form. In our exercise, the original equation represents a degenerate conic and simplifies to a single line when factored.

Graphing is a visual way to comprehend the nature of a conic section, providing insights into its slope, direction, and points of intersection with the axes. Even in degenerate cases, graphing the resultant lines or points can help us understand the underlying geometric relationships.

Standard Form of a Conic

The standard form of a conic is an equation simplified to make the classification of the curve clear. For non-degenerate conics, standard forms help identify the center, vertices, foci, axes, and other features of the curve.

An example of a standard form for a quadratic conic is \(ax^2+bxy+cy^2+dx+ey+f=0\)). The degenerate conic from our exercise equates to \(x^2-10xy+y^2=0\)), and after factoring, it doesn't fit the mold of a traditional conic section. Instead, it represents a pair of overlapping lines, revealing that the standard form can vary depending on the nature of the curve. Recognizing and working with different standard forms is a foundational aspect of studying conic sections.