Problem 35 - Algebra 2

Question

Write an equation for the circle that satisfies each set of conditions. center at \((4,2),\) tangent to \(x\) -axis

Step-by-Step Solution

Answer

The equation is \((x - 4)^2 + (y - 2)^2 = 4.\)

Step 1: Understand the Circle Equation

The general equation for a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Here, the center \((h, k)\) is given as \((4, 2)\). Therefore, the equation should look like \((x - 4)^2 + (y - 2)^2 = r^2\).

Step 2: Identify the Tangent Condition

Since the circle is tangent to the \(x\)-axis, this means the distance from the center \((4, 2)\) to the \(x\)-axis is equal to the radius. The distance from the center to the \(x\)-axis is \(|2 - 0| = 2\). Thus, \(r = 2\).

Step 3: Substitute the Radius into the Equation

Using the radius found in step 2, we substitute\(r = 2\) into the circle equation:\[(x - 4)^2 + (y - 2)^2 = 2^2\] which simplifies to: \[(x - 4)^2 + (y - 2)^2 = 4.\]

Key Concepts

Radius of a Circle

Understanding the radius of a circle is fundamental in geometry. In the context of a circle's equation, the radius is the distance from the circle's center to its edge. Knowing the radius helps us form the correct equation for the circle. This distance is represented by "\(r\)" in the standard circle equation:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here, \(\sqrt{r^2}\) gives the actual radius. In practical terms, the radius is crucial because it determines the size of the circle. In this specific exercise, seeing that the circle is tangent to the x-axis greatly simplifies finding \(r\). The center's y-coordinate is 2, which is the distance (or radius) to the x-axis, making our radius 2. This understanding about radius is key to writing the circle's equation accurately.If the radius were miscalculated, the equation would lead to a completely different circle. Therefore, accurate determination is essential.

Center of a Circle

The center of a circle is an important location point defined by two coordinates, \((h, k)\). These numbers represent how far the circle is from the origin along the x and y-axis. In the standard circle equation

• \((x - h)^2 + (y - k)^2 = r^2\)

\(h\) and \(k\) are the center coordinates. This understanding helps correctly plot the circle on a coordinate plane.For our problem, the center is given as \((4, 2)\). This means the circle is:

• 4 units away from the y-axis
• 2 units away from the x-axis

The center is a fixed point from which all points on the circle are equidistant. The information about the center allows us to know where the circle is located and is used to write the circle equation.

Tangent to x-axis

When a circle is tangent to the x-axis, it means the circle touches the x-axis at exactly one point. This property of tangency gives us specific insights about both the circle and its equation.The tangency condition tells us that the circle's radius is exactly equal to the y-coordinate of the center, \(k\). Since the circle doesn't cross the axis but just touches it, the distance from the center to the x-axis is the radius.In our case, the center

• \((4, 2)\)

means the circle's radius must be 2. This condition ensures that the circle touches but does not pass through the x-axis. This insight helps determine the radius right away, simplifying the process of forming the equation for the circle. Understanding tangency helps solve and verify circular problems swiftly and accurately.