Problem 51 - Precalculus with Limits

Question

In Exercises 51-56, sketch (if possible) the graph of the degenerate conic. \(y^2-16x^2=0\)

Step-by-Step Solution

Answer

The graph of this degenerate conic consists of a pair of lines that intersect at the origin. The lines have slopes of 4 and -4, indicating that they slope upward and downward, respectively. The lines are defined by the equations \(y=4x\) and \(y=-4x\).

Step 1: Rewrite the Equation in a More Familiar Form

The given equation is \(y^2-16x^2=0\), but it can be helpful to rewrite this in the form \(a^2-b^2=(a-b)(a+b)\). Here, \(a\) is \(y\), and \(b\) is \(4x\). Rewriting gives: \(y^2-(4x)^2=0\) or \(y-4x)(y+4x)=0\).

Step 2: Determine the Zeroes

To help in sketching, find the zeroes of the equation. This can be done by solving for \(y\) when \(x=0\) and vice versa. Solving \(y-4x=0\) for \(y\) when \(x=0\) gives \(y=0\). Solving \(y+4x=0\) for \(y\) when \(x=0\) also gives \(y=0\). Therefore, the line crosses the y-axis at the origin. For the x-intercept, plug \(y=0\) into the equations gives \(x=0\). Therefore, the lines cross at the origin.

Step 3: Sketching the Graph

Using the information obtained in the first two steps, sketch the two lines on graph paper. Both lines pass through the origin, with the line for \(y-4x=0\) sloping upward and the line for \(y+4x=0\) sloping downward. The slope of a line is determined by \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) for coordinates \((x_1,y_1)\) and \((x_2,y_2)\). Here \(m=4\) for \(y=4x\) and \(m=-4\) for \(y=-4x\).

Key Concepts

Equations of Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone, resulting in ellipses, parabolas, and hyperbolas. A unique case occurs when the conic "degenerates" into two intersecting lines. This happens when certain conditions are met in the equation. A general second-degree equation for a conic is given by \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). By manipulating this equation, one can identify the type of conic section or recognize it as degenerate.

In this exercise, the equation \( y^2 - 16x^2 = 0 \) is a classic form of a degenerate conic. Here, the zero constant expands to factored linear terms: \((y - 4x)(y + 4x) = 0\). Solving these factorized parts separately gives two linear equations: \( y - 4x = 0 \) and \( y + 4x = 0 \), indicating that our degenerate conic is essentially two intersecting lines.

Graphing Techniques

Graphing conic sections requires understanding the shape and position dictated by their equations. For degenerate conics, like the equation \( y^2 - 16x^2 = 0 \), the graph is founded on linear equations derived from factorization.

Here, the equations \( y = 4x \) and \( y = -4x \) are linear, making graphing straightforward.

Steps to graph:

• Start by plotting the origin, as both lines intersect here.
• Draw the line \( y = 4x \) by moving through points like (1, 4) since the slope \( m = 4 \).
• Draw the line \( y = -4x \) using points like (1, -4), as the slope \( m = -4 \).

This results in an 'X' shape, with both lines intersecting at the origin, illustrating the nature of the degenerate conic.

Slope of Lines

The slope of a line is a measure of its steepness, defined by \( m = \frac{y_2-y_1}{x_2-x_1} \). This concept is pivotal in analyzing linear components of degenerate conics.

For the lines \( y = 4x \) and \( y = -4x \), we determine their slopes as \( m = 4 \) and \( m = -4 \) respectively.

Understanding slope:

• The positive slope \( m = 4 \) indicates the line rises to the right.
• The negative slope \( m = -4 \) shows it falls to the right.
• These slopes define the direction and angle at which the lines cross axes.

Knowing how to calculate and interpret slopes allows for accurate graphing and understanding of the lines forming a degenerate conic.