Problem 51 - 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; no points.

Step-by-Step Solution

Answer

Draw a circle and a horizontal line outside its boundary.

Step 1: Understanding the Problem

We need to draw a line and a circle such that they do not intersect at any point. This implies the line must be completely outside the circle, either above, below, or beside it.

Step 2: Visualizing Possible Positions

Consider the circle centered at a point, say the origin for simplicity, with some radius. A line can be horizontal, vertical, or diagonal, but it must not pass through or touch the circle. Imagining the plane, position the line entirely outside the circle’s boundary.

Step 3: Choosing Specific Equations

To sketch this situation, consider a circle centered at the origin: \(x^2 + y^2 = r^2\). Choose \(y = mx + c\) for the line. To ensure no intersection, choose \(c > r\) or \(c < -r\) for a horizontal line (\(m = 0\)).

Step 4: Drawing the Sketch

1. Draw the circle centered at the origin with a chosen radius, say radius 3.2. Choose a horizontal line, such as \(y = 4\), ensuring \(c > r\), where \(r = 3\).3. The line is above the circle and does not intersect it.

Key Concepts

Circle Equation

The equation of a circle is a foundational concept in geometry that describes all the points equidistant from a given center point. For a circle centered at the origin (0,0), the equation is expressed in its simplest form as \(x^2 + y^2 = r^2\). Here, \(r\) represents the radius, which is the distance from the center to any point on the circle's perimeter.
A circle's general equation can be written as \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) represent the circle's center coordinates. Understanding this equation helps in graphing a circle by locating its center and drawing a curve with the specified radius.
Key points to remember:

• The radius must always be positive.
• The center of the circle can be any point on a coordinate plane.
• Changing the value of \(r\) affects the size of the circle but not its shape.

Line Equation

A line in a two-dimensional plane can be defined using the line equation \(y = mx + c\), where \(m\) represents the slope, and \(c\) is the y-intercept. This equation is known as the slope-intercept form, a very convenient method to describe linear equations.
The slope \(m\) indicates the steepness of the line and its direction. If \(m>0\), the line inclines upwards; if \(m<0\), the line declines downwards; and if \(m=0\), the line is horizontal. The y-intercept \(c\) tells us where the line crosses the y-axis.
Essential features:

• A larger absolute value of \(m\) means a steeper line.
• A positive \(c\) places the line above the origin, while a negative \(c\) places it below.
• Horizontal lines have a slope of zero, making their equation simply \(y = c\).

Intersection of Graphs

Graphing the intersection of a circle and a line involves checking where, if at all, the two graphs meet on the coordinate plane. This is where solutions of the two equations are the same (i.e., points that simultaneously satisfy both equations).
For a line and a circle, determining their intersection involves:

• Setting up the equations: a circle \(x^2 + y^2 = r^2\) and a line \(y = mx + c\).
• Substituting the line equation into the circle equation to find common solutions.
• Simplifying and solving for \(x\) and \(y\) to identify intersection points.

In scenarios where the line and circle should not intersect, as in our exercise, we ensure \(c > r\) or \(c < -r\) for a horizontal line. This keeps the line outside the circle, with no shared points, ensuring no intersections on the graph.