Problem 86 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &\frac{(x-2)^{2}}{36}+\frac{(y+2)^{2}}{25} \leq 1\\\ &\frac{(x+1)^{2}}{9}+\frac{(y-3)^{2}}{25} \leq 1 \end{aligned}$$

Step-by-Step Solution

Answer

Graph both ellipses, then shade the region where their interiors overlap.

Step 1: Identify the type of conic sections

The given inequalities are in the form of an ellipse, represented by \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) where \((h, k)\) is the center, and \(a\) and \(b\) are the semi-major and semi-minor axes respectively.

Step 2: Analyze the first ellipse

For the first inequality \( \frac{(x-2)^2}{36} + \frac{(y+2)^2}{25} \leq 1 \), the center is \((2, -2)\). It has a semi-major axis of 6 (because \(\sqrt{36} = 6\), greater than \(\sqrt{25}\)) along the x-axis and a semi-minor axis of 5 along the y-axis.

Step 3: Analyze the second ellipse

For the second inequality \( \frac{(x+1)^2}{9} + \frac{(y-3)^2}{25} \leq 1 \), the center is \((-1, 3)\). It has a semi-major axis of 5 along the y-axis and a semi-minor axis of 3 along the x-axis.

Step 4: Graph the first ellipse

On a coordinate plane, plot the center at \((2, -2)\). From this point, draw an ellipse that extends 6 units to the right and left along the x-axis and 5 units up and down along the y-axis. Shade the interior of the ellipse to represent the solution set.

Step 5: Graph the second ellipse

On the same coordinate plane, plot its center at \((-1, 3)\). From this point, draw an ellipse that extends 3 units to the right and left along the x-axis and 5 units up and down along the y-axis. Shade the interior of this ellipse as well to represent the solution set.

Step 6: Identify the overlapping region

The overall solution set of the system of inequalities is where the interiors of the two ellipses overlap. This region represents the set of all \((x, y)\) points that satisfy both inequalities.

Key Concepts

Ellipses

Ellipses are a fascinating type of conic section. They appear as oval shapes on a plane. They can be cleverly defined through an algebraic equation. In general form, an ellipse is given by the equation \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Here, \( (h, k) \) represents the center of the ellipse. The values \( a \) and \( b \) denote the lengths of the semi-major and semi-minor axes, respectively.

• The semi-major axis is the longest diameter of the ellipse.
• The semi-minor axis is the shortest diameter.
• If \( a > b \), the ellipse stretches along the x-axis. If \( b > a \), it stretches along the y-axis.

Understanding ellipses involves recognizing their symmetry and navigating their axes. Graphically, they help visualize constraints in systems like inequalities. So, when you see equations of ellipses, think about distinct oval paths on a plane, each specified by its center and its axes.

Graphing Solutions

Graphing solutions to systems of inequalities, like the ones involving ellipses, is a hands-on process. It reveals the set of all possible solutions visually on a coordinate plane. The solution is often represented by shading regions. To graph elliptic inequalities, begin by plotting their centers. Use the ellipse equations to decide how to stretch the figure along the axes.

• For example, consider \( \frac{(x-2)^2}{36} + \frac{(y+2)^2}{25} \leq 1 \). The center is \( (2, -2) \), with a semi-major axis of 6 along the x-axis and a semi-minor axis of 5 along the y-axis.
• Next, draw the ellipse and shade the area within to indicate the solution set.

Repeat this for any further inequalities, and look for intersections. The overlapping shaded region gives the combined solution, satisfying more than one inequality. This technique is vital for systems of inequalities as it visually illustrates all feasible solutions.

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. Of these, ellipses and circles are closely related as they both signify closed curves.

• Circles are a special type of ellipse where \( a = b \).
• Hyperbolas differ as they split into two distinct open curves.
• Parabolas open in a single direction and appear as U-shaped or inverted-U-shaped curves.

Conic sections arise in various real-world contexts, from orbits of planets to architectural designs. Their study helps understand geometric shapes and forms through algebraic equations.In systems of inequalities, conic sections define boundaries for regions of possible solutions. Recognizing and graphing these sections help us comprehend feasible solution areas related to complex systems. By exploring them, you understand how simple algebraic equations translate into significant geometric paradigms.