Problem 58 - Precalculus: Mathematics for Calculus

Question

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. $$\left\\{\begin{array}{l}\frac{x^{2}}{16}+\frac{y^{2}}{9}=1 \\\\\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\end{array}\right.$$

Step-by-Step Solution

Answer

Intersection points are \( \left(\frac{12}{5}, \frac{12}{5}\right), \left(-\frac{12}{5}, -\frac{12}{5}\right), \left(\frac{12}{5}, -\frac{12}{5}\right), \left(-\frac{12}{5}, \frac{12}{5}\right) \).

Step 1: Understand the Equation of Ellipses

The first equation \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \) is an ellipse centered at the origin with semi-major axis 4 (along the x-axis) and semi-minor axis 3 (along the y-axis). The second equation \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \) is also an ellipse centered at the origin but with semi-major axis 4 (along the y-axis) and semi-minor axis 3 (along the x-axis).

Step 2: Set the Equations Equal to Solve for Intersection Points

To find intersection points, set both ellipses' equations equal. Equate the two equations since they equal 1: \[ \frac{x^2}{16} + \frac{y^2}{9} = \frac{x^2}{9} + \frac{y^2}{16} \]Simplify this equation to find common solutions for \(x\) and \(y\).

Step 3: Simplify the Equation

Multiply through by the least common multiple of denominators, which is 144, to clear the fractions: \[ 9x^2 + 16y^2 = 16x^2 + 9y^2 \]Rearrange terms to isolate terms involving \(x^2\) and \(y^2\): \[ 7y^2 = 7x^2 \] This simplifies to: \[ y^2 = x^2 \]

Step 4: Solve for Variables

From \( y^2 = x^2 \), we find: \( y = x \) or \( y = -x \).Substitute these into one of the original ellipse equations.

Step 5: Find Intersection Points for \( y = x \)

Substituting \( y = x \) into \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \), we have: \( \frac{x^2}{16} + \frac{x^2}{9} = 1 \). Combine terms using a common denominator: \[ \frac{9x^2 + 16x^2}{144} = 1 \]\[ 25x^2 = 144 \] Solve for \(x\): \[ x^2 = \frac{144}{25} \] \[ x = \pm\frac{12}{5} \]Thus, \( y = \pm\frac{12}{5} \). An intersection occurs at \( \left(\frac{12}{5}, \frac{12}{5}\right) \) and \( \left(-\frac{12}{5}, -\frac{12}{5}\right) \).

Step 6: Find Intersection Points for \( y = -x \)

Substituting \( y = -x \) into \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \), we have: \( \frac{x^2}{16} + \frac{x^2}{9} = 1 \). Following similar steps in Step 5, we've already found the solution, so the intersection points are \( \left(\frac{12}{5}, -\frac{12}{5}\right) \) and \( \left(-\frac{12}{5}, \frac{12}{5}\right) \).

Step 7: Summary and Graph Sketching

Now that we have the intersection points: \( \left(\frac{12}{5}, \frac{12}{5}\right), \left(-\frac{12}{5}, -\frac{12}{5}\right), \left(\frac{12}{5}, -\frac{12}{5}\right), \left(-\frac{12}{5}, \frac{12}{5}\right) \), sketch the graph.Draw two ellipses centered at the origin — one elongated horizontally, the other vertically. Mark the intersection points on the graph.

Key Concepts

Ellipse Equations

Understanding the equations of ellipses is the first step to solving intersection problems like this. Each given ellipse equation represents a specific shape on the coordinate plane with unique properties.
The standard form for an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). In this form, \(a\) and \(b\) represent the semi-major and semi-minor axes. For instance, the first ellipse equation \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \) shows that the ellipse has a semi-major axis of 4 along the x-axis and a semi-minor axis of 3 along the y-axis.
Similarly, the second ellipse \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \) has a major orientation switched to the y-axis, with semi-major axis 4 and semi-minor axis 3. The horizontal and vertical stretches change how wide and tall the ellipse is. This knowledge helps when switching axes and solving for intersection points.

Graph Sketching

Sketching graphs of ellipses gives visual aid to understanding intersections. For the given ellipses, start by plotting the axes. Then draw each ellipse with their properties.

• Ellipse 1: The equation \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \) tells us the ellipse is wider horizontally, extending 4 units along the x-axis and 3 units along the y-axis.
• Ellipse 2: Similarly, the equation \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \) presents an ellipse taller vertically, extending 4 units along the y-axis and 3 units along the x-axis.

Each coordinate is labeled for accuracy. Once the basic shapes are outlined, mark the intersection points identified in previous steps. These points are \( \left(\pm \frac{12}{5}, \pm \frac{12}{5}\right) \), which visually show where and how the ellipses overlap.

Coordinate Geometry

Coordinate geometry helps in analyzing equations to find shared solutions. For ellipses, these solutions are the intersecting points. The equations provided are:

• \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \)
• \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \)

Using algebra, set these equal and find \(x\) and \(y\) values that satisfy both. Multiply by a common denominator to eliminate fractions, getting a straightforward equation like \( y^2 = x^2 \). This indicates two possibilities: \(y = x\) or \(y = -x\). Substitute these into one ellipse equation and solve for exact points. This confirms the theoretical equation derived.

Conic Sections

Ellipses are part of a family of shapes known as conic sections. This group includes circles, parabolas, and hyperbolas, each defined by the intersection of a plane with a cone. Ellipses specifically occur when the plane intersects the cone at an angle, forming a closed curve. Recognizing these properties assists in understanding shape characteristics like axes orientation and areas of overlap.
Solving ellipse intersection problems involves examining where these closed curves meet on the plane. This practical application of conic sections highlights key geometry principles and improves visual spatial skills.