Problem 62 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the points of intersection of \(x^{2}+4 y^{2}=20\) and \(x+2 y=6\).

Step-by-Step Solution

Answer

The points of intersection are \((4, 1)\) and \((2, 2)\).

Step 1: Express one variable in terms of the other from the second equation

We start with the linear equation: \( x + 2y = 6 \). Let's express \( x \) in terms of \( y \):\[x = 6 - 2y\]

Step 2: Substitute into the first equation

Substitute \( x = 6 - 2y \) into the ellipse equation \( x^2 + 4y^2 = 20 \):\[(6 - 2y)^2 + 4y^2 = 20\]

Step 3: Expand and simplify the equation

Expand \((6 - 2y)^2\) and substitute it into the equation:\[(6-2y)^2 = 36 - 24y + 4y^2\]Substitute and simplify:\[36 - 24y + 4y^2 + 4y^2 = 20\]\[8y^2 - 24y + 36 = 20\]

Step 4: Solve the quadratic equation for y

Rearrange the equation:\[8y^2 - 24y + 16 = 0\]Divide throughout by 8:\[y^2 - 3y + 2 = 0\]Factorize:\[(y - 1)(y - 2) = 0\]Solve for \( y \): \( y = 1 \) or \( y = 2 \).

Step 5: Find corresponding x-values

Substitute \( y = 1 \) back into \( x = 6 - 2y \):\[x = 6 - 2(1) = 4\]Substitute \( y = 2 \) back into \( x = 6 - 2y \):\[x = 6 - 2(2) = 2\]

Step 6: Write the points of intersection

The points of intersection are \((4, 1)\) and \((2, 2)\).

Key Concepts

Intersection of Curves

Finding the intersection of curves is a common problem in calculus that involves determining the exact point(s) where two graphs meet on a coordinate plane. In our exercise, we are tasked with finding where the ellipse \(x^2 + 4y^2 = 20\) intersects with the line \(x + 2y = 6\).

• To solve such problems, we typically express one variable in terms of another using one of the given equations.
• Substituting this expression into the other equation gives a single equation with one variable, making it easier to solve.
• The solutions to this equation identify potential intersection points.
• We verify these points by substituting back to check if they satisfy both original equations.

This technique can be applied to many problems and is a fundamental skill in mathematics.

Quadratic Equations

A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our example, after substitution and simplification, we derived a quadratic equation for \(y\): \(y^2 - 3y + 2 = 0\).

• Quadratic equations can often be solved by factoring, completing the square, or using the quadratic formula.
• In this problem, the equation was factored into \((y - 1)(y - 2) = 0\), so the solutions are \(y = 1\) and \(y = 2\).
• Finding these values allows us to determine corresponding \(x\)-values by back-substitution.

Quadratic equations are pivotal in understanding various dynamic systems and modeling different scenarios in science and engineering.

Simultaneous Equations

Simultaneous equations are sets of equations with multiple variables that are solved together because they share these variables. For our problem, we used simultaneous equations to find points of intersection. We started with:

• \(x + 2y = 6\)
• \(x^2 + 4y^2 = 20\)

Solving simultaneous equations allows us to find a mutual solution for all included equations. By manipulating one equation to express \(x\) in terms of \(y\), we substituted into the other:
- Bringing it down to a single equation for solving, simplifies the process significantly.- The obtained solutions then must satisfy both equations to confirm they are points of intersection.
Understanding how to handle simultaneous equations is essential not just in geometry but in economics, engineering, and statistical data analysis.

Ellipse and Line

An ellipse is a symmetrical curved shape, the general equation of which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). In our example, the equation is given as \(x^2 + 4y^2 = 20\), which describes an ellipse centered at the origin.A line, on the other hand, is represented by the equation \(y = mx + c\) or any equivalent form like \(x + 2y = 6\).

• Intercepting an ellipse with a line involves finding points that satisfy both the line and the ellipse equations.
• These intersection points reflect where the line "cuts" through the ellipse, which can be zero, one, or two points depending on their configuration.

Understanding these intersections helps in physics, astronomy, and graphic design where such shapes dynamically interact.