Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 31-32: Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$ \left\\{\begin{array}{l} x^{2}+4 y^{2}=36 \\ x^{2}+y^{2}=12 \end{array}\right. $$

Step-by-Step Solution

Answer

Intersection points are (±4, ±2).

Step 1: Set up the System of Equations

We are given two equations: 1. \( x^{2} + 4y^{2} = 36 \) 2. \( x^{2} + y^{2} = 12 \). The goal is to find the points \((x, y)\) that satisfy both equations simultaneously.

Step 2: Solve the System Algebraically by Subtraction

Subtract the second equation from the first equation to eliminate \( x^{2} \):\[(x^{2} + 4y^{2}) - (x^{2} + y^{2}) = 36 - 12 \4y^{2} - y^{2} = 24 \3y^{2} = 24\]Solve for \( y^{2} \) by dividing both sides by 3:\[ y^{2} = 8 \]Take the square root of both sides:\[ y = ext{either } oxed{ ext{positive or negative }rac{2 ext{ } imes ext{ } ext{sqrt(2)}}{2}} \].

Key Concepts

Intersection Points

When working with a set of equations, one of the main goals is often to find the points where the graphs of the equations meet, known as intersection points. These points are significant because they satisfy both equations simultaneously.

To find intersection points, start by solving the system of equations simultaneously. In our case, the equations are:

• \( x^2 + 4y^2 = 36 \)
• \( x^2 + y^2 = 12 \)

Solving these algebraically, as demonstrated in the original step-by-step solution, reveals the specific \( (x, y) \) coordinates that lie on both geometric shapes represented by these equations. In simpler terms, the intersection points are where the ellipse (from the first equation) and the circle (from the second equation) overlap on the graph.

Understanding the concept of intersection points not only helps us in mathematics, but it's also a key concept in fields like physics and engineering, providing solutions to real-world intersection problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to study geometry using a coordinate plane. This powerful tool helps us visualize abstract algebraic equations and their relationships. In this exercise, the coordinate plane becomes a canvas where shapes take form, and their intersections are painted as points of interest.

Both equations provided,

• \( x^2 + 4y^2 = 36 \)
• \( x^2 + y^2 = 12 \)

are visualized on a coordinate plane. The first equation, representing an ellipse, and the second, representing a circle, showcase how different mathematical entities can coexist in the same space. Plotting them on a graph allows you to see where these distinct shapes intersect.

Skills in coordinate geometry are essential, not only for solving textbook problems but also for creating designs in computer graphics and in fields such as architecture and engineering. In various applications, it allows precise modeling of locations, distances, and areas.

Algebraic Solutions

Algebraic solutions involve manipulating equations to find the desired values. This process includes operations such as substitution, elimination, and factoring. Solving systems of equations algebraically is a critical skill in mathematics as it bypasses the need for graphical methods, instead relying on pure computation.

In our system of equations, subtraction is used to eliminate \( x^2 \), leaving us with:

• \( 4y^2 - y^2 = 24 \)
• \( 3y^2 = 24 \)

From here, solving for \( y^2 \) gives us \( y^2 = 8 \). Subsequently, by taking the square root, we find the values of \( y \). This method is efficient and precise, serving as a backbone for solving many complex mathematical problems.

Mastering algebraic techniques is fundamental for students as they progress to higher levels of mathematics, where these methods become indispensable tools for tackling more intricate equations, such as those involving multiple variables or higher-degree polynomials.

Ellipses

Ellipses are fascinating geometric figures that look like elongated circles. The equation \( x^2 + 4y^2 = 36 \) describes an ellipse when visualized on a coordinate plane. Unlike circles that have a single radius, ellipses have two radii: the semi-major and semi-minor axes.

For the given equation, the ellipse is centered at the origin \((0,0)\). This is typically the case when the equation takes the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, the denominator of \( x^2 \) and \( y^2 \) reflects the relative lengths of each axis. The presence of 4 in \( 4y^2 \) implies that the ellipse is stretched along the y-axis.

Ellipses are not just mathematical curiosities; they have applications in physics (such as planetary orbits) and engineering (like gear design). Knowing how to identify and work with ellipses is crucial for students in understanding more complex, real-world dynamic systems.