Problem 60 - 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. $$\left\\{\begin{aligned}25 x^{2}+144 y^{2} &=3600 \\\144 x^{2}+25 y^{2} &=3600\end{aligned}\right.$$

Step-by-Step Solution

Answer

The intersection points are \((6, 3), (-6, 3), (6, -3), (-6, -3), (0, 9), (0, -9)\).

Step 1: Understanding the Equation of Ellipses

The given pair of equations represent ellipses. The general equation for an ellipse centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This means we can transform the given equations into standard form to understand their dimensions.

Step 2: Convert the Equations into Standard Form

Divide each equation by 3600 to convert them into standard form: \[\frac{25x^2}{3600} + \frac{144y^2}{3600} = 1\] simplifies to \(\frac{x^2}{144} + \frac{y^2}{25} = 1\).\[\frac{144x^2}{3600} + \frac{25y^2}{3600} = 1\] simplifies to \(\frac{x^2}{25} + \frac{y^2}{144} = 1\).

Step 3: Identify Properties of Each Ellipse

The first ellipse, \(\frac{x^2}{144} + \frac{y^2}{25} = 1\), has a semi-major axis of 12 along the x-axis and a semi-minor axis of 5 along the y-axis. The second ellipse, \(\frac{x^2}{25} + \frac{y^2}{144} = 1\), has a semi-major axis of 12 along the y-axis and a semi-minor axis of 5 along the x-axis. This helps in sketching them as their axes are swapped.

Step 4: Solve the System of Equations

To find intersections, solve the system by substitution or elimination. From \(x^2/144 + y^2/25 = 1\) and \(x^2/25 + y^2/144 = 1\), make \(x^2\) and \(y^2\) the subject: Equation 1 gives \(x^2 = 144(1 - y^2/25)\).Equation 2 gives \(x^2 = 25 - 25y^2/144\).Equate the derived equations to solve for \(y\), result in values for \(y\) and substitute back to find corresponding \(x\) values.

Step 5: Solve for Coordinates

Solve the derived equations to find that for \(y = \pm 3\), \(x = \pm 6\), and for \(y = \pm 9\), \(x = \pm 0\). Thus, the intersection points are \((6, 3), (-6, 3), (6, -3), (-6, -3)\), and points where ellipses coincide at \((0, 9), (0, -9)\).

Step 6: Verify and Sketch Graphs

Substitute the intersection points back into the original equations to verify solutions. Sketch both ellipses on the same coordinate plane: one elongated horizontally and the other vertically. Label the intersection points clearly on the graph.

Key Concepts

Ellipse Standard Form

The ellipse standard form is a way to write the equation of an ellipse so that we can easily understand its shape and dimensions. In the center of the coordinate plane, an ellipse can be written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This equation shows

• \(a\): the semi-major axis (longest radius),
• \(b\): the semi-minor axis (shortest radius),
• If \(a > b\), the ellipse stretches horizontally;
• if \(b > a\), it stretches vertically.

To convert any given ellipse equation to standard form, you divide every term in the equation by a constant, so that the right side equals one. This transformation simplifies identifying the specific geometry of the ellipse by clearly showing the lengths of the axes. Ensuring the equation is in standard form helps make any calculations or graphing straightforward and accurate.

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. To find the solution to such a system, which in this case is determining where the two ellipses intersect, we solve the equations simultaneously. There are several methods to achieve this:

• **Substitution:** Solve one equation for a variable, then substitute into the other equation.
• **Elimination:** Adjust equations so adding or subtracting eliminates a variable, simplifying the system.

For our ellipse equations \(\frac{x^2}{144} + \frac{y^2}{25} = 1\) and \(\frac{x^2}{25} + \frac{y^2}{144} = 1\), solving them can reveal the points where they intersect by finding the common \(x\) and \(y\) values. This technique is fundamental to coordinate geometry when dealing with multiple shapes.

Graphing Ellipses

Graphing ellipses is an essential skill to visualize their shapes and positions. This process begins by using the standard form of each ellipse. Start by identifying the lengths of the semi-major and semi-minor axes from the equation. Once these are known, follow these steps:

• Plot the center point (the origin in standard form).
• From the center, measure out in both the x-direction and y-direction according to the axes' lengths.
• Draw the ellipse smoothly around these point stretches, ensuring symmetry around both axes.

For the example ellipses, one stretches more along the x-axis while the other along the y-axis. Plotting them on the same coordinate grid helps you find visually where they overlap or cross each other, confirming the algebraic results from solving the system.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use algebra to study geometry. It involves using a coordinate plane to find the positions of points, and to understand shapes through equations. In dealing with ellipses:

• We describe their positions and dimensions using algebraic equations.
• We solve equations to find specific points of interest, like intersection points between two ellipses.

Coordinate geometry provides a great advantage because it connects algebraic methods to geometrical figures, making it easier to both solve and visualize problems like finding intersections. This helps bridge understanding across different mathematical disciplines, especially in understanding complex interactions between shapes like ellipses.