Problem 41 - Contemporary Precalculus

Question

Identify the conic section and use technology to graph it. $$4 x^{2}+y^{2}+24 x-4 y+36=0$$

Step-by-Step Solution

Answer

The given equation represents a point, not a conic section, after rewriting it in the form $4 (x+3)^2 + (y-2)^2=0$. The point is (-3, 2). To graph it, use a graphing calculator or an online graphing utility like Desmos or GeoGebra and plot the point (-3, 2) on the graph.

Step 1: Rewrite the equation

First, let's rewrite the given equation in standard form. To do this, we'll complete the square for both x and y terms: $$4 (x^2+6x) + (y^2-4y)=-36$$ Notice the pattern (a+b)^2 = (a^2 + 2ab + b^2). For the x terms, a = x and 2ab = 6x. So, we need to figure out what b is: $$b^2 = \frac{6^2}{4} = 9 \Rightarrow b = 3$$ For the y terms, a = y and 2ab = -4y. So, we need to figure out what b is: $$b^2 = \frac{(-4)^2}{4} = 4 \Rightarrow b = -2$$ Now we can rewrite the equation after completing the square: $$4 (x+3)^2 + (y-2)^2=-36 + 4(3^2) + (2^2)$$ $$4 (x+3)^2 + (y-2)^2=0$$

Step 2: Determine the conic section

Now that the equation is in a more recognizable form, we can determine the conic section it represents. The equation represents an ellipse if both squared terms have coefficients with different signs and the right side of the equation is positive. It represents a circle if both squared terms have the same coefficient and the right side of the equation is positive. In our case, both squared terms have positive coefficients, but the right side of the equation is equal to 0. This means the equation represents a point, not a conic section. The point is given by the center of the would-be ellipse or circle, so the point is (-3, 2).

Step 3: Graph it using technology

Finally, use a graphing tool or calculator to plot the point (-3, 2). The most common tool for this would be a graphing calculator or an online graphing utility like Desmos or GeoGebra. To plot the point on a graphing calculator or an online graphing utility: 1. Open the calculator or utility. 2. Input the equation $$(x+3)^2 + (y-2)^2=0$$. Some utilities might accept the original equation $$4 x^{2}+y^{2}+24 x-4 y+36=0$$ as well. 3. View the plot. The point (-3, 2) should be visible on the graph, representing the given equation.

Key Concepts

Completing the Square

Completing the square is a technique used in algebra to transform quadratic expressions into a form that is easier to analyze. This form typically reveals geometric properties or helps in solving equations.
To complete the square:

• Take the quadratic term and the linear term. Look at the coefficient of the linear term (say, 6x).
• Divide this coefficient by 2 and then square the result: \[\left( \frac{6}{2} \right)^2 = 9\]
• Add and subtract this square within the expression. This method converts \( x^2 + 6x \) into \( (x+3)^2 \), which is easier to handle.

By completing the square in the exercise, we transformed the quadratic equation into a conic section form. Understanding this transformation is critical in proceeding to identify and analyze shapes like circles and ellipses.

Ellipse

An ellipse is a kind of conic section that looks like a stretched circle. It is defined by its major and minor axes, which are the longest and shortest diameters respectively.
Typically, an ellipse can be represented in its standard form:\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]Here, \((h, k)\) represents the center of the ellipse, and \(a\) and \(b\) represent the distances from the center to the ellipse along the x and y-axis respectively.
In the given exercise, after manipulating the equation to complete the square, we discovered that instead of an ellipse, the equation simplifies to the equation of a single point. This happens because both squared terms had equal values and they summed up to zero. Therefore, the potential ellipse degenerated into a single point at (-3, 2).

Graphing Technology

Graphing technology is a crucial tool in visualizing mathematical concepts like conic sections. It allows students and teachers to plot equations accurately and explore the resulting figures.

Popular tools include:

• Graphing calculators like TI-84 or TI-Nspire
• Online platforms, such as Desmos or GeoGebra
• Software packages like MATLAB or Mathematica

To use these tools effectively: - Input the algebraic equation directly, whether it is in its original form or a transformed version after completing the square. - Observe the plotted graph to identify key features like points, lines, or curves. In the exercise, after transforming the equation, it becomes clear that the graph would depict a single point, which these tools can pinpoint easily. Understanding how to input and manipulate equations in graphing technology will enhance your mathematical proficiency.

Algebraic Solution

Algebraic solutions involve manipulating and simplifying equations to identify and understand their properties without necessarily relying on visualization tools.
This approach is fundamental because it relies on theoretical understanding and reduces errors in interpretation.

• Analyzing the structure of the equation helps identify what kind of conic section it might represent.
• Completing the square lets us reveal this structure, making it easier to interpret or solve the equation.
• From the simplified equation form, we concluded that there was no traditional ellipse, but instead, it represented a point.

The algebraic manipulation helps lead students step-by-step from a complex equation to a form that summarizes the key properties of the equation as evidenced in the exercise provided, ultimately enabling us to conclude that the equation in question doesn't describe a full-scale elliptic shape but rather a point.