Problem 17 - A Graphical Approach to College Algebra

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 is \((x + 3)^2 + (y + 2)^2 = 4\).

Step 1: Understand the center-radius form of a circle

The equation of a circle in center-radius form is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) represents the center of the circle and \(r\) is the radius.

Step 2: Identify the center of the circle

The problem states that the center of the circle is \((-3, -2)\). Therefore, \(h = -3\) and \(k = -2\).

Step 3: Determine the radius using tangent condition

Since the circle is tangent to the \(x\)-axis, its distance from the \(x\)-axis is equal to the radius. The \(y\)-coordinate of the center is \(-2\), so the distance (radius) is the absolute value of that, which is \(|-2| = 2\).

Step 4: Write the equation of the circle

Now that we have the center \((-3, -2)\) and the radius \(r = 2\), we plug these values into the center-radius form of the equation of a circle: \((x + 3)^2 + (y + 2)^2 = 2^2\). This simplifies to \((x + 3)^2 + (y + 2)^2 = 4\).

Key Concepts

center-radius form

The center-radius form is a fundamental concept when dealing with circles in a coordinate plane. This equation is one of the most efficient ways to represent a circle mathematically. The formula is given by \[(x - h)^2 + (y - k)^2 = r^2\]where:

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

This form is crucial because it directly highlights two key pieces of information—the position of the circle's center and its size through the radius. The equation states that for any point \((x, y)\) on the circle, the distance from the center \((h, k)\) will always equal the radius \(r\). Thus, this form is used to easily manipulate and analyze circles from given conditions.
Understanding the center-radius form simplifies solving problems involving circles such as determining tangents or plotting the circle in Cartesian coordinates.

tangent to axis

When we say a circle is tangent to an axis, we mean it touches that axis at exactly one point without crossing it. In our example, the circle is tangent to the \(x\)-axis, which implies that the radius is equal to the vertical distance from the center of the circle to the \(x\)-axis.
Consider the center of our circle, \((-3, -2)\). The \(y\)-coordinate here is \(-2\), and because the circle is tangent to the \(x\)-axis, the circle must just touch the axis at a height of \(2\) units above or below, depending on the quadrant the circle is located in.

• This touching point occurs precisely when: \[k + r = 0 \]
• or when \[k - r = 0 \]

This condition eliminates the possibility of the circle cutting the axis anywhere, ensuring a unique touchpoint. It's a significant takeaway because it informs us how to handle the circle's position relative to axes when only given partial data.

radius determination

Understanding how to determine the radius of a circle is paramount, especially when given conditions such as being tangent to an axis. The radius is simply the shortest distance from the center to the circle's boundary.
In cases where the circle is tangent to the \(x\)-axis, this distance becomes evident through the circle's \(y\)-coordinate.- The distance from the center of the circle to the \(x\)-axis is simply the absolute value of the \(y\)-coordinate of the center.- Given the center at \((-3, -2)\), the radius, therefore, is \[|k| = |-2| = 2\]The radius calculation is vital as it allows for the complete specification of the circle's equation and confirms geometrical properties like tangency.
Understanding this relationship ensures you can formulate the correct equation of a circle even when only partial positional or dimensional criteria are provided.