Problem 31 - Algebra and Trigonometry

Question

\(23-34\) Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+16=4\left(y^{2}+2 x\right) $$

Step-by-Step Solution

Answer

The equation represents a degenerate hyperbola.

Step 1: Simplify the equation

Start with the given equation: \(x^{2} + 16 = 4(y^{2} + 2x)\).Distribute the 4 on the right side:\[ x^{2} + 16 = 4y^{2} + 8x \]Rearrange the terms:\[ x^{2} - 8x + 16 - 4y^{2} = 0 \].

Step 2: Complete the square for \(x\)

Focus on the \(x\)-terms: \(x^{2} - 8x\).To complete the square, take the coefficient of \(x\), which is \(-8\), divide by 2 and square it:\[ \left( \frac{-8}{2} \right)^2 = 16 \]Now, rewrite the \(x\)-terms:\[ (x - 4)^2 - 16 \]Substitute back into the equation:\[ (x - 4)^2 - 16 + 16 = 4y^{2} \]The equation simplifies to:\[ (x - 4)^2 = 4y^{2} \].

Step 3: Identify the conic section

Compare the equation \((x - 4)^2 = 4y^{2}\) to the standard conic equations.This equation resembles the form of a hyperbola:\[ (x - h)^2 - (ky)^2 = 0 \]This configuration represents a degenerate hyperbola which may not graph as usual.

Step 4: Conclusion about graph existence

Since \((x - 4)^2 = 4y^{2}\) appears as a degenerate hyperbola, this implies that in a typical Cartesian plane, it might not map to meaningful visual features like vertices or asymptotes on the graph and might reduce to intersecting lines or a single line.

Key Concepts

Completing the Square

Completing the square is a valuable technique used to transform quadratic expressions into a perfect square trinomial. This procedure is crucial for identifying the type of conic section an equation represents. To complete the square in an expression like \(x^2 - 8x\):

• Take the linear coefficient, which is \(-8\).
• Divide it by 2, yielding \(-4\).
• Then, square it to get \(16\).

This turns the expression into \((x - 4)^2 - 16\). This transformation is essential as it allows us to recognize and manipulate the equation more effortlessly. Completing the square simplifies the quadratic expression so we can identify the conic section more effectively, aiding in further analysis or graphing tasks.

Hyperbola

A hyperbola is one of the four types of conic sections. It is formed when a plane cuts through both nappes of a double cone. Hyperbolas have a distinctive shape with two separate curves or branches.In the equation \((x - 4)^2 = 4y^2\), we recognize elements of a hyperbola's structure but quickly observe that it is not typical. Standard hyperbola forms, like \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), help identify its features such as:

• Center, \((h, k)\)
• Vertices, derived from \(a\)
• Foci, spaced further from the center using \(c\) where \(c^2 = a^2 + b^2\)
• Asymptotes guiding the curve's branches

The given equation is described as a degenerate hyperbola, implying irregular or minimal graph features.

Degenerate Conic

A degenerate conic occurs when a conic section doesn't form a typical curve but reduces to simpler sets like points or lines. Understanding this concept is key for graphing irregular conic sections.The original equation, \((x - 4)^2 = 4y^2\), leads us into the realm of degenerate conics.

• If simplified, it suggests reducing geometric structures, often intersecting lines rather than standard hyperbola curves.
• Such outcomes arise when compared to regular equations and fail to plot as expected conic sections.

Recognizing a degenerate conic enables one to know why it might not produce a discernible graph and sets the stage for understanding more complex sections in algebra.

Graphing Conic Sections

Graphing conic sections is about visual representation of equations into planes. Each type of conic—circle, ellipse, parabola, or hyperbola—has specific features for plotting. For hyperbolas, usually, parameters such as center, vertices, and asymptotes guide the graph. However, in degenerate cases like our example, such plots are often either nonexistent or atypical.

• Degenerate conics might graph, at most, as intersecting lines, or no graph.
• This behavior changes standard expectations for conic visual plots.

Understanding how and when to graph conics effectively is crucial for mathematics, especially when dealing with degenerate forms. Thus, always think of checking the equation forms and refer to standard conic formulas to decide feasible graph features.