Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((-3,-2) ;\) tangent to the \(x\) -axis (Hint: "tangent to" means touching at one point.)

Step-by-Step Solution

Answer

The center-radius form of the circle is \((x + 3)^2 + (y + 2)^2 = 4\).

Step 1: Identify Center and Point of Tangency

The center of the circle is given as \((-3, -2)\). The hint indicates the circle is tangent to the \(x\)-axis. This means the distance from the center to the \(x\)-axis, which is the radius of the circle, is equal to the absolute value of the \(y\)-coordinate of the center; i.e., \(|-2| = 2\).

Step 2: Calculate the Radius

Since the circle is tangent to the \(x\)-axis, the shortest distance from the center \((-3, -2)\) to the \(x\)-axis is equal to the radius. Therefore, the radius \(r\) of the circle is 2.

Step 3: Write the Center-Radius Form of the Circle

The center-radius form of a circle is given by the equation \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Substituting the center \((-3, -2)\) and the radius \(2\) into the formula, we get \((x + 3)^2 + (y + 2)^2 = 2^2\).

Step 4: Simplify the Equation

Simplifying the right side of the equation \((x + 3)^2 + (y + 2)^2 = 2^2\), results in \((x + 3)^2 + (y + 2)^2 = 4\). This is the center-radius form of the circle.

Key Concepts

Center-Radius Form of a Circle

The center-radius form is a specific way to express the equation of a circle. It highlights the circle's center and its radius. We use the formula

• \((x - h)^2 + (y - k)^2 = r^2\)

Here,

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius of the circle.

The center-radius form is very useful. It directly shows the location and size of the circle. For example, if the circle's center is at \((-3, -2)\) and the radius is 2, the equation is given by:

• \((x + 3)^2 + (y + 2)^2 = 4\)

This shows that any point \((x, y)\) on the circle satisfies this equation.

Understanding Tangency to an Axis

When a circle is tangent to one of the coordinate axes, it means the circle touches that axis at exactly one point. This property is essential for determining the circle's radius in specific problems.For the circle tangent to the \(x\)-axis, as in our example, the distance from the center to the axis is precisely the circle's radius. Since the center is \((-3, -2)\), the distance or radius here is the absolute value of the \(y\)-coordinate, which is 2.Due to this tangential relationship, one can easily figure out the radius of the circle simply by knowing the coordinates of the center and the axis to which the circle is tangent. In this way, tangency provides crucial information about the circle's geometry and allows us to solve problems swiftly.

Steps for Radius Calculation

Finding the radius of a circle when it is tangent to an axis involves understanding the shortest distance from the center to that axis.### Calculate the RadiusTo find the radius of a circle tangent to the \(x\)-axis:1. **Identify the circle's center.** - Given as \((-3, -2)\).2. **Check the axis of tangency**: - Circle is tangent to the \(x\)-axis.3. **Use the relevant coordinate to determine the radius**: - Since it is tangent to the \(x\)-axis, use the \(y\)-coordinate. The radius \(r\) is \[|y| = |-2| = 2\]This simple relationship makes it straightforward to determine the circle's radius when tangency is involved. The calculation can be extended to cases where a circle is tangent to the \(y\)-axis, then it would use the \(x\)-coordinate instead.