Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Draw a sketch of the two graphs described with the indicated number of points of intersection. A line and a circle; no points

Step-by-Step Solution

Answer

The line should be positioned completely outside the circle, parallel to the x-axis, ensuring no points of intersection occur.

Step 1: Understanding the Problem

To solve this problem, we need to draw a line and a circle on a graph such that they do not intersect at any point. This means the line can either be above, below, or tangent to the circle without touching it.

Step 2: Drawing the Circle

Start by drawing the circle. Choose a center and a radius. Let's say the circle is centered at the origin \((0,0)\) with a radius of \(r\). This will give us the equation \(x^2 + y^2 = r^2\). On the graph, draw a circle using these parameters.

Step 3: Positioning the Line

Now we want to draw a line that does not intersect the circle. If the circle is centered at the origin \( (0,0) \) with radius \( r \,\), draw the line with equation \(y = r + 1\). This line will be above the circle and accordingly not touch or intersect with it. Ensure that this line runs parallel to the x-axis.

Step 4: Verifying No Intersection

Check that the line drawn does not intersect the circle. For this line, substitute \(y = r + 1\) into the circle's equation \(x^2 + y^2 = r^2\). You will have \(x^2 + (r + 1)^2 = r^2\). This simplifies to \(x^2 = -2r - 1\), which has no real solution, confirming there are no intersection points.

Key Concepts

Graphing

Graphing is a powerful visual tool in geometry and algebra that helps us understand the relationships between different equations and shapes. It transforms equations into visual representations, allowing us to easily see intersections, slopes, and other interactions. When graphing a circle and a line, our goal might be to observe whether or not these shapes intersect.

Here are some simple steps to graph a circle and a line:

• Identify the parameters of the equations, such as the center and radius for a circle, or the slope and y-intercept for a line.
• Sketch the circle on the graph, ensuring the accuracy of its size and position.
• Position the line based on its equation, observing its relationship with the circle for intersections or tangents.

Graphing is not just about plotting points; it's about understanding space and relationships between mathematical objects. This graphical approach is crucial when determining non-intersecting or tangent scenarios.

Equation of a circle

The equation of a circle is a fundamental concept in geometry. It provides a precise mathematical description of a circle's size and positioning on a plane. A circle is defined by its center and radius, and its equation takes the form:

• y^2 - 2hy + h^2 + x^2 - 2kx + k^2 = r^2

where \(h, k\) is the center of the circle, and \(r\) is its radius.

For a circle centered at the origin \(0,0\), this simplifies to \(x^2 + y^2 = r^2\). This equation indicates that the distance from any point \(x, y\) on the circle to the center \(0,0\) is always \(r\). When positioning a line around such a circle, we ensure the line does not cross or touch the circle's boundary, confirming a non-intersecting relationship.

Equation of a line

The equation of a line is another essential tool for creating and analyzing graphs. In its simplest form, a line can be expressed using the slope-intercept form:

• y = mx + b

where \(m\) represents the slope and \(b\) is the y-intercept. The slope \(m\) describes the steepness of the line, while the intercept \(b\) specifies where the line crosses the y-axis.

When we need a line that does not intersect with a circle, we consider the relative positions of the two shapes. For instance, a line such as \(y = r + 1\), parallel to the x-axis above the circle's topmost point, will never touch or intersect a circle centered at the origin with radius \(r\). Such an understanding of line placement relative to circles is vital for solving graphing problems.