Problem 70 - Precalculus Essentials

Question

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned}(x-3)^{2}+(y+1)^{2} &=9 \\\y &=x-1\end{aligned}$$

Step-by-Step Solution

Answer

The intersection points of the circle equation and the line equation are at (2,1) and (4,3).

Step 1: Graphing the equations

First, graph the given circle equation \((x-3)^{2}+(y+1)^{2} =9\) and the line equation \(y = x-1\) on the same rectangular coordinate system. You will notice two points where the line crosses the circle. These are the intersection points.

Step 2: Solve the equations

Now, solve the equations simultaneously to get the exact values of the intersection points. Substitute \(y = x-1\) into the circle equation, so you have \((x-3)^{2}+(x-1+1)^{2} =9\). After solving, you get \(x = 2, 4 \). Substitute \(x = 2, 4\) into the equation \(y = x-1\) to find y. So the solutions are (2,1) and (4,3).

Step 3: Verify the solutions

Finally, verify that these points satisfy both the equations. Substitute the points into both the circle equation and the line equation, you will find that they satisfy both the equations.

Key Concepts

Understanding Circle Equations

When analyzing a circle on a coordinate plane, the circle equation plays a critical role. It is generally expressed in the form \((x-h)^2 + (y-k)^2 = r^2\). Here, the point \((h, k)\) represents the center of the circle, while \(r\) is the radius. This form enables us to easily identify the circle's characteristics, such as its position and size.

• Center: The circle in the equation \((x-3)^{2}+(y+1)^{2} =9\) has a center at \((3, -1)\), obtained by comparing with the general form \((h, k)\).
• Radius: The radius \(r\) is the square root of 9, which is 3, indicating how far each point on the circle is from its center.

Understanding the structure of a circle equation helps in sketching it on a graph, making it easier to analyze how other geometrical figures, like lines, interact with the circle. Identifying these features swiftly highlights the potential points of intersection with any line.

Decoding Line Equations

A line equation, usually in the form \(y = mx + c\), represents all the points that form a straight line on a plane. The slope, \(m\), indicates the line's steepness, while \(c\) represents the y-intercept, where the line crosses the y-axis.

In the line equation \(y = x-1\):

• Slope \(m = 1\): This indicates a 45-degree angle of rise, moving one unit up for every unit moved right.
• Y-intercept \(c = -1\): This tells us the line crosses the y-axis at \(-1\).

Graphing this type of equation helps visualize its path across the coordinate system. When paired with another equation, like a circle equation, it allows us to determine critical intersections visually or algebraically by solving the equations simultaneously. This essential application of line equations is pivotal in problem-solving contexts, such as finding intersection points with circles.

Finding Intersection Points

Finding intersection points between a line and a circle involves determining the points that satisfy both equations simultaneously. These points are where the line crosses the circle on a graph.

To find the intersection points, we substitute the expression for \(y\) from the line equation \(y = x-1\) into the circle equation \((x-3)^{2}+(y+1)^{2} =9\). This substitution leads to a quadratic equation in terms of \(x\): \((x-3)^{2}+(x-1+1)^{2} =9\), or \((x-3)^{2} + x^{2} = 9\).

Solving this equation gives the \(x\)-values of the intersection points, which are \(x = 2\) and \(x = 4\). By substituting these \(x\)-values back into the line equation, we find the corresponding \(y\)-values, resulting in the points (2,1) and (4,3).

Lastly, verifying these points by plugging them back into both the circle and line equations ensures their validity. Doing so confirms that both (2,1) and (4,3) lie at the intersection of the given line and circle, providing an effective way to solve geometry problems involving systems of equations.