Problem 46 - 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

The coordinates of intersection points are (0, ±3) and (±4, 0).

Step 1: Identify the Equations of the Ellipses

We have two equations representing two ellipses:1) \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \)2) \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \). The first equation represents an ellipse with its major axis along the x-direction, and the second one has its major axis along the y-direction.

Step 2: Set Up the System to Find Intersection Points

To find the intersection points of the ellipses, we need to solve this system of equations simultaneously.Given:\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]\[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \].

Step 3: Solve Using Subtraction of Equations

Subtract the second equation from the first:\[ \left( \frac{x^2}{16} + \frac{y^2}{9} \right) - \left( \frac{x^2}{9} + \frac{y^2}{16} \right) = 0 \].Simplify to find:\[ \frac{x^2}{16} - \frac{x^2}{9} + \frac{y^2}{9} - \frac{y^2}{16} = 0 \].

Step 4: Simplify and Combine Terms

Factor the equation step by step:- Convert subtraction into a common denominator.- Combine the \(x^2\) terms: \( \frac{x^2}{144} (-9 + 16) = \frac{7x^2}{144} \).- Combine the \(y^2\) terms: \( \frac{y^2}{144} (16 - 9) = \frac{7y^2}{144} \).The equation becomes:\[ \frac{7x^2}{144} = \frac{7y^2}{144} \].

Key Concepts

Ellipses

An ellipse is a geometric shape that looks like an elongated circle or an oval. It is defined by two main axes: the major axis and the minor axis. The major axis is the longest diameter of the ellipse, while the minor axis is the shortest.
When discussing ellipses in a coordinate plane, we often deal with standard equation forms. The general form for an ellipse centered at the origin is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.
For the given problem, we have two specific ellipses:

• The first ellipse, \(\frac{x^2}{16} + \frac{y^2}{9} = 1\), has a major axis along the x-direction. The semi-major axis length is 4 (since \( \sqrt{16} = 4\)) and the semi-minor axis length is 3 (since \( \sqrt{9} = 3\)).
• The second ellipse, \(\frac{x^2}{9} + \frac{y^2}{16} = 1\), has its major axis along the y-direction. This means its semi-major axis length is 4, and its semi-minor axis length is 3.

Understanding these properties helps visualize how each ellipse stretches along the x or y-axis.

System of Equations

Solving a system of equations involves finding values for variables that satisfy all equations in the system. In the context of finding intersection points of ellipses, we are dealing with a system of two equations.
For this exercise, the system consists of:

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

To find solutions common to both equations, we need to solve them simultaneously. This typically means using techniques such as substitution, elimination, or graphical analysis.
In this case, the subtraction method is employed where equation 2 is subtracted from equation 1 to simplify the terms and make it easier to solve for intersection points. This process involves careful handling of fractions and ensuring terms are aligned for combination.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use algebraic equations to represent geometric shapes on a coordinate plane. This field of study bridges the gap between algebra and geometry, making it easier to visualize and solve geometric problems.
When we work with the intersection of ellipses, coordinate geometry helps us visualize these curves on a plane. The coordinate axes serve as reference points for plotting ellipses and determining where they intersect.

• Graphs of both ellipses are plotted based on their respective equations.
• The points where these graphs meet are the solutions to the system of equations: they represent the intersection points of the ellipses.

By sketching the graphs of these equations, we can see not just where they intersect but also gain insight into their relative positions and orientations.