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. $$ 9 x^{2}+4 y^{2}+72 x-16 y+160=0 $$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Rearrange the Equation

Start by rearranging the given equation grouping the x and y terms together. We have: \[ 9x^2 + 72x + 4y^2 - 16y = -160 \] This allows us to focus on each variable separately for completing the square.

Step 2: Complete the Square for x

Look at the terms involving x: \(9x^2 + 72x\). Factor out the 9 from these terms:\[ 9(x^2 + 8x) \]To complete the square, take half of the coefficient of x, square it, and add/subtract inside the parentheses. Half of 8 is 4 and 4 squared is 16. So we have:\[ 9(x^2 + 8x + 16 - 16) = 9((x + 4)^2 - 16) = 9(x + 4)^2 - 144 \]

Step 3: Complete the Square for y

Look at the terms involving y: \(4y^2 - 16y\). Factor out the 4:\[ 4(y^2 - 4y) \]To complete the square, take half of the coefficient of y, square it, and add/subtract inside the parentheses. Half of -4 is -2 and -2 squared is 4. So we have:\[ 4(y^2 - 4y + 4 - 4) = 4((y - 2)^2 - 4) = 4(y - 2)^2 - 16 \]

Step 4: Substitute and Simplify

Substitute the completed squares back into the equation.\[ 9(x + 4)^2 - 144 + 4(y - 2)^2 - 16 = -160 \] Combine and simplify the constants on one side of the equation:\[ 9(x + 4)^2 + 4(y - 2)^2 = 0 \] Add 160 to both sides: \[ 9(x + 4)^2 + 4(y - 2)^2 = 160 \]

Step 5: Identify the Conic

For a conic section of the form \( Ax^2 + By^2 = C \) with both A and B positive, the equation represents an ellipse. Since our simplified equation,\[ 9(x + 4)^2 + 4(y - 2)^2 = 160 \] fits this form (both coefficients are positive), it is an ellipse.

Key Concepts

Completing the Square

Completing the square is a handy mathematical technique that transforms quadratic equations into a form that is easier to solve or analyze. This method is particularly useful when working with conic sections, like ellipses, as it allows us to rewrite the equation in a more familiar and workable form. To complete the square for a term like \( x^2 + bx \), follow these steps:

• Identify the coefficient of \( x \), which is \( b \).
• Divide \( b \) by 2.
• Square the result to get the number that completes the square.
• Add and subtract this number within the expression.

This results in an expression such as \( (x + \, ext{number})^2 \), making it much simpler to handle within an equation. By adopting this process, we can convert messy expressions into neat squares, allowing us to easily identify the conic section they describe.

Ellipse

An ellipse is a beautiful and symmetric shape that resembles a stretched circle. It is one of the four types of conic sections, which are the curves obtained by slicing a cone with a plane. The general equation of an ellipse in a standard form looks like \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where:

• \( (h, k) \) is the center of the ellipse.
• \( a \) and \( b \) represent the lengths of the semimajor and semiminor axes respectively.

Ellipses display unique properties, like having two focal points such that the sum of the distances from any point on the ellipse to the foci is constant. This characteristic makes ellipses a key concept in geometry and physics. In our exercise, the equation \( 9(x + 4)^2 + 4(y - 2)^2 = 160 \) was identified as an ellipse because both terms containing \( x \) and \( y \) were positive, a defining characteristic of ellipses.

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, typically written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Quadratics are fundamental in algebra, providing the basis for solving various mathematical problems and modeling real-world scenarios.In the context of conic sections, quadratic equations help describe curves like ellipses, parabolas, and hyperbolas. Solving these equations often involves strategies like factoring, using the quadratic formula, or completing the square. Knowing these methods allows us to manipulate equations into forms that we can easily interpret and use.In our exercise, the initial equation \( 9x^2 + 4y^2 + 72x - 16y + 160 = 0 \) involved completing the square to isolate and identify it as an ellipse, showcasing the power and utility of quadratic equations in mathematics.