Problem 53 - A Graphical Approach to College Algebra

Question

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a circle; two points.

Step-by-Step Solution

Answer

The line and circle can intersect at points \((2.5, 4.5)\) and \((-4.5, -2.5)\).

Step 1: Understanding the Problem

We are tasked with drawing a line and a circle such that they intersect at exactly two distinct points. Our goal is to sketch these graphs while ensuring the correct number of intersections.

Step 2: Choosing a Circle

First, we choose a circle. Let's consider a simple circle centered at the origin with a radius of 5. The equation of the circle will be \(x^2 + y^2 = 25\).

Step 3: Choosing a Line

Next, we choose a line that will intersect the circle in two points. A line that passes through the circle but not through its center can achieve this. For simplicity, let's consider the line given by the equation \(y = x + 2\).

Step 4: Finding Points of Intersection

Substitute the line equation into the circle equation: \(x^2 + (x+2)^2 = 25\). Expand and simplify: \(x^2 + x^2 + 4x + 4 = 25\), which simplifies to \(2x^2 + 4x - 21 = 0\).

Step 5: Solving the Quadratic Equation

Solve the quadratic equation \(2x^2 + 4x - 21 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 2\), \(b = 4\), and \(c = -21\). After solving, we get \(x = 2.5\) and \(x = -4.5\).

Step 6: Calculating y-values

Substitute the x-values back into the line equation \(y = x + 2\) to find the corresponding y-values. For \(x = 2.5\), \(y = 4.5\); for \(x = -4.5\), \(y = -2.5\).

Step 7: Sketching the Graphs

Draw the circle centered at the origin with a radius of 5. Then, draw the line \(y = x + 2\), making sure it intersects the circle at \((2.5, 4.5)\) and \((-4.5, -2.5)\), thus confirming two points of intersection.

Key Concepts

Intersections

In geometry, an intersection is a fundamental concept that refers to the point or points where two or more geometric shapes meet or cross each other. When it comes to graphing a line and a circle, understanding where these graphs intersect is crucial. In our example, these intersections occur at exactly two points. To determine intersections:

• Identify the equations that represent the geometric shapes you are working with. For this example, a circle and a line.
• Substitute one equation into the other to find common solutions. This step gives you potential x-values where the two graphs intersect.
• Compute the corresponding y-values using the found x-values to get the full points of intersection.

Once you have your points of intersection, you can accurately sketch or plot these points on a graph, confirming where and how each shape crosses the other. Understanding intersections is key to analyzing how graphs relate to one another in a coordinate plane.

Circle Equation

The equation of a circle is a key concept in geometry. It represents all the points that make up a circle in the coordinate plane. The general equation of a circle with a center at oordinate (platedisc)xyxy\((h, k)\) h and a radius \(r\) is given by:\[ (x - h)^2 + (y - k)^2 = r^2 \]For a circle centered at the origin, oordinate ky is simply \(x^2 + y^2 = r^2\). This form makes it easy to calculate or represent circles in a graph by analyzing their radius and origin.nula. In our example, we use the equation\(x^2 + y^2 = 25\)which indicates a circle centered at the origin \( (0,0) \),with a radius of 5 (since \(r^2 = 25\), so \(r = 5\)).
Understanding the equation of the circle helps in identifying its geometric representation and aids in finding where other shapes, like lines, may intersect it.

Line Equation

Lines are one of the simplest curves in algebra and geometry, commonly represented by linear equations. The standard form of a line equation is expressed as \(y = mx + b\),where \(m\) is the slope, indicating the steepness and direction of the line, and \(b\) is the y-intercept, specifying where the line crosses the y-axis.Consider the line given by the equation\(y = x + 2\).Here:

• The slope \(m\) is 1, meaning the line rises one unit up for each unit it moves right on the graph.
• The y-intercept \(b\) is 2, showing where the line intersects the y-axis.

Finding the intersection points of this line with a circle's equation involves substituting the line equation into the circle's equation and solving for x and y. Mastering line equations is key to understanding how to graph linear relationships and identify intersections with other shapes.