Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(4,-1) \text {, tangent to the } x \text {-axis } $$

Step-by-Step Solution

Answer

The equation of the circle is \((x-4)^2 + (y+1)^2 = 1\).

Step 1: Understand Circle Basics

A circle is defined by its center and its radius. The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Identify Circle Center

From the problem, the center of the circle is given as \( C(4, -1) \). This implies that the parameters \( h \) and \( k \) in the circle's equation are \( h = 4 \) and \( k = -1 \).

Step 3: Identify Tangency Condition

The circle is tangent to the x-axis. This means the distance from the center of the circle to the x-axis is equal to the radius \( r \). Since the center is at \( (4, -1) \), the distance to the x-axis is \(|-1| = 1\), which is the radius \( r \) of the circle.

Step 4: Write the Circle Equation

Using the circle's center \((4, -1)\) and the radius \( r = 1 \), substitute these values into the circle's equation format: \[(x-4)^2 + (y+1)^2 = 1^2\]

Key Concepts

Circle Center and Radius

In a circle, the center and the radius are fundamental properties. The center of a circle is the fixed point from which every point on the circle is equidistant. It acts as the circle's anchor. The radius is the distance from the center to any point on the circle. These properties help in defining the equation of a circle, which is: \[(x-h)^2 + (y-k)^2 = r^2\] Here,

• are the coordinates of the circle’s center.
• r is the radius.

For instance, if the center is at \((4, -1)\) and the radius is 1, we substitute , , and . The equation becomes: \[(x-4)^2 + (y+1)^2 = 1\]. Understanding where the Center \(C\) and fit in the equation is key to solving circle problems.

Tangency Condition

The tangency condition is critical when a circle is tangent to a line, such as the x-axis. **Tangent** means the circle just touches the line at one single point. For a circle with center \((h, k)\) to be tangent to the x-axis, the radius must equal the absolute value of the of the center. This is because the perpendicular distance from the center of the circle to the line is the radius when tangent conditions are met. In our case, the center is \((4, -1)\). This means the distance from the center to the x-axis is <|-1|>, which equals 1. This distance is used as the radius . This condition ensures that as you move downward along the , the circle's circumference just touches the x-axis.

Distance to X-axis

The distance from the center of a circle to the x-axis plays a significant role, especially when determining tangency. To find this distance, you simply take the absolute value of the of the circle's center. For a center \((h, k)\), where is -1, the distance from the center to the x-axis becomes:\[|k| = |-1| = 1\]

• This distance is the shortest path from the center vertically to the x-axis.
• It establishes the length of the radius when the circle is tangent to the axis.

In simpler terms, if you draw a line perpendicularly from the center to the x-axis, the length of this line is exactly what defines the radius under tangency conditions.