Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples \(3-5) .\) 9 x^{2}+4 y^{2}+72 x-16 y+160=0

Step-by-Step Solution

Answer

The given equation represents a degenerative conic, specifically a single point.

Step 1: Separate and Rearrange Terms

The given equation is \( 9x^2 + 4y^2 + 72x - 16y + 160 = 0 \). First, rearrange the equation grouping the \(x\) terms and \(y\) terms: \( 9x^2 + 72x + 4y^2 - 16y = -160 \).

Step 2: Factor Coefficients of the Quadratic Terms

Factor out the coefficients of \(x^2\) and \(y^2\) from the grouped terms: \( 9(x^2 + 8x) + 4(y^2 - 4y) = -160 \).

Step 3: Complete the Square for x-terms

For the \(x\)-terms \(x^2 + 8x\), complete the square by adding and subtracting \( (\frac{8}{2})^2 = 16 \) inside the parenthesis: \( 9(x^2 + 8x + 16 - 16) = 9((x+4)^2 - 16) \).

Step 4: Complete the Square for y-terms

For the \(y\)-terms \(y^2 - 4y\), complete the square by adding and subtracting \((\frac{-4}{2})^2 = 4\) inside the parenthesis: \( 4(y^2 - 4y + 4 - 4) = 4((y-2)^2 - 4) \).

Step 5: Rewrite the Equation in Completed Square Form

Substitute back the completed squares: \[ 9((x+4)^2 - 16) + 4((y-2)^2 - 4) = -160 \]. Simplify: \[ 9(x+4)^2 - 144 + 4(y-2)^2 - 16 = -160 \]. Combine constants: \[ 9(x+4)^2 + 4(y-2)^2 = 0 \].

Step 6: Analyze the Equation

The form \( Ax^2 + By^2 = 0 \) with \(A\) and \(B\) both positive indicates this represents a degenerate conic, specifically a single point. Here, it collapses to the point where both squares are zero.

Key Concepts

Completing the Square

Completing the square is a method used to transform quadratic equations into a perfect square trinomial. This procedure simplifies the equation and reveals important properties of the conic section.
It involves adding and subtracting the appropriate term to make the expression a perfect square.
Here are the steps:

• Identify the quadratic term and the linear term.
• Take half of the linear coefficient, square it, and both add and subtract this value within the equation.
• Factor the resulting trinomial as a squared binomial, incorporating any necessary constants outside the square.

For example, in the exercise, to complete the square for the term \(x^2 + 8x\), we added and subtracted 16 to form \((x+4)^2\). This formed a perfect square trinomial, essential for rewriting the equation in a form that reveals its characteristics.

Degenerate Conic

In conic sections, a degenerate conic is a special case where the conic form "collapses". This means that instead of forming a full cone shape like an ellipse or a hyperbola, the result is a simpler geometric entity.
These can be points, lines, or intersecting lines.
In our exercise, the final form of the equation was \(9(x+4)^2 + 4(y-2)^2 = 0\). Here, both terms squared must equal zero, meaning:

• \((x+4)^2 = 0\), which simplifies to \(x = -4\).
• \((y-2)^2 = 0\), which simplifies to \(y = 2\).

This situation indicates a degenerate conic, where the entire conic reduces to a single point, specifically \((-4, 2)\). This illustrates how algebraic manipulation can completely redefine the nature of an equation.

Equation Rearrangement

Rearranging an equation is crucial for clarity especially when dealing with conic sections. By grouping like terms, the equation becomes easier to manage and solve. Rearrangement is typically performed in preparation for methods like completing the square.
The exercise initially had the equation \(9x^2 + 4y^2 + 72x - 16y + 160 = 0\). By rearranging this to separate \(x\) and \(y\) terms, it turned into \(9(x^2 + 8x) + 4(y^2 - 4y) = -160\). This step simplifies the process of completing the square for both \(x\) and \(y\) variables.
The strategic rearrangement not only aids in simplification but also provides clarity, allowing us to see where transformations or factorizations need to take place within the equation. Being comfortable with rearrangement opens the door to efficiently employing various mathematical techniques.