Problem 43 - Contemporary Precalculus

Question

Identify the conic section and use technology to graph it. $$9 x^{2}+25 y^{2}-18 x+50 y=191$$

Step-by-Step Solution

Answer

In summary, the given equation represents an ellipse centered at (1, -1) with semi-major axis √(397/225) and semi-minor axis 3. To graph the ellipse, a student can use graphing software or calculators such as Desmos, GeoGebra, or TI-84 by entering the equation in its standard form and adjusting the viewing scale as needed.

Step 1: Rewrite the equation in standard form

First, we need to gather the terms with the same variables together and complete the squares for both x and y. $$9x^2 - 18x + 25y^2 +50y = 191$$ Divide by 9 for x terms and 25 for y terms: $$(x^2 - 2x) + (y^2+2y)=\frac{191}{225} $$ Now we'll complete the given squares: \((x^2 - 2x + 1) + (y^2 +2y+1) =\frac{191}{225}+1+1\) Now the equation becomes: $$(x-1)^2+\frac{1}{9}(y+1)^2 = \frac{397}{225}$$

Step 2: Identify the conic section

Now we can identify the conic section by looking at the standard form equation. For an ellipse, the equation has the following form: $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$ Comparing our equation to this form, we find that: - \(h=1\), \(k=-1\) - \(a^2= \frac{397}{225}\) - \(b^2= 9\) So the given equation is an ellipse centered at \((1,-1)\) with semi-major axis \(a = \sqrt{\frac{397}{225}}\) and semi-minor axis \(b= 3\).

Step 3: Graph the ellipse using technology

To graph the ellipse, we can use graphing software or calculators like Desmos, GeoGebra, or TI-84. Follow these steps: 1. Open the preferred graphing tool. 2. Enter the equation in the standard form: \(\frac{(x-1)^2}{\frac{397}{225}}+\frac{(y+1)^2}{9}=1\). 3. Adjust the graph's scale, if necessary, to see the complete ellipse. 4. Observe the ellipse that is centered at (1, -1) with semi-major axis \(\sqrt{\frac{397}{225}}\) and semi-minor axis 3. With these steps, the student should be able to identify the given conic section as an ellipse and graph it using technology.

Key Concepts

Ellipse

An ellipse is a type of conic section that represents a closed curve. It's like a stretched circle with two axes: the major and minor axes. Imagine it as an oval shape on paper. The center of an ellipse is at point \(h, k\), and the lengths of its semi-major and semi-minor axes are \(a\) and \(b\) respectively. Here’s what makes the ellipse unique:

• It has two foci (plural of focus) inside the ellipse along the major axis.
• All points on the ellipse have the same total distance to the two foci.
• If \(a > b\), the ellipse is longer horizontally; if \(b > a\), it's longer vertically.

In our equation example, the ellipse is centered at (1, -1). This center position helps in easily graphing and understanding the ellipse's placement on a coordinate plane. Knowing how to identify an ellipse by its equation is critical for graphing and understanding these beautiful shapes.

Completing the Square

Completing the square is an essential algebraic technique used to convert quadratic equations into a neater form. This method helps to transform an equation into a perfect square trinomial, which makes solving and graphing easier. Let's break it down:

• First, you rearrange the terms to group the \(x\) terms and \(y\) terms separately.
• For each group, focus on creating a perfect square trinomial by editing the equation: typically done by adding and subtracting a number inside the group to form \( (x \, \pm \, m)^2 \) or \( (y \, \pm \, n)^2 \).
• The added numbers weight the equation on one side, so equivalent values are included on the other to maintain balance.

In our problem, completing the square for both \(x\) and \(y\) terms gives us the standard form necessary for identifying the ellipse and drawing it accurately. Understanding this method aids in visualizing how algebraic adjustments impact graphed equations.

Graphing Technology

Graphing technology, such as Desmos, GeoGebra, or graphing calculators, simplifies the process of visualizing mathematical equations. These tools perform excellent jobs in displaying graphs once equations are provided:

• They accept mathematical inputs and automatically generate visual representations.
• You can easily manipulate the scale to better view graphs, such as our ellipse.
• Graphing tools offer instant feedback, helping check accuracy and understand features like the center or axes lengths.

Using these technologies, students can plug the standard form equation of the ellipse directly to see its shape on the plane. Letting technology do this rigorous task allows students to focus on comprehending the core concepts and seeing real-time changes when they adjust equations.

Standard Form Equation

The standard form equation is crucial in categorizing conic sections, such as circles, ellipses, parabolas, and hyperbolas. For an ellipse, the standard equation looks like:\[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\]Here's what each part represents:

• \((h, k)\) determines the center of the ellipse, giving you a starting point for graphing.
• \(a^2\) and \(b^2\) relate directly to the lengths of the semi-major and semi-minor axes, respectively. \(a\) aligns with the longer axis, and \(b\) with the shorter.
• If the equation equals 1, it conforms to the normalized form that makes graphing straightforward.

Our task was to rewrite the given equation into this standard form. By completing the square and readjusting, we successfully reduced the expression, aiding in the identification of the ellipse and its features. Understanding this form serves as the backbone for recognizing and analyzing conic sections.