Problem 52 - Algebra and Trigonometry

Question

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ x^{2}+(y-2)^{2}=4 $$

Step-by-Step Solution

Answer

Center: (0,2); Radius: 2

Step 1: Identify the Circle Equation

The equation of the circle is given as \( x^2 + (y - 2)^2 = 4 \). This is in the standard form \((x - h)^2 + (y - k)^2 = r^2\) where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Determine the Center of the Circle

Comparing the given equation with the standard form, we see that \( h = 0 \) and \( k = 2 \). Thus, the center of the circle is \((0, 2)\).

Step 3: Calculate the Radius of the Circle

The equation \( x^2 + (y - 2)^2 = 4 \) shows that \( r^2 = 4 \). By taking the square root, the radius \( r = \sqrt{4} = 2 \).

Step 4: Sketch the Graph of the Circle

To sketch the graph, plot the center \((0, 2)\). Draw a circle with radius 2 around this center. Ensure that every point on the circle is 2 units away from the center.

Key Concepts

Center of the Circle

In the world of geometry, the term "center of the circle" refers to a specific point that is equidistant from every point along the circle's circumference. This point is crucial because it determines the position of the circle in the coordinate plane.
In a standard circle equation of the form \[ (x - h)^2 + (y - k)^2 = r^2 \]the center is denoted by the coordinates \((h, k)\). To identify the center, simply extract these values from the equation.

• \(h\) is the x-coordinate of the center.
• \(k\) is the y-coordinate of the center.

In our example equation, \[ x^2 + (y - 2)^2 = 4 \]it can be seen that \(h = 0\) and \(k = 2\). Therefore, the center of the circle is at the point \((0, 2)\). This means that every point on our circle is centered around this point on the graph.

Radius of the Circle

The radius of a circle is the distance from its center to any point on its edge. This simple yet essential concept defines a circle's size.In the circle's standard form, \[ (x - h)^2 + (y - k)^2 = r^2 \]the term \(r^2\) represents the square of the radius.
To find the actual radius, take the square root of \(r^2\).
In the given example, \[ x^2 + (y - 2)^2 = 4 \]the equation shows \(r^2 = 4\). By taking the square root, we find that the radius \(r = \sqrt{4} = 2\).

• Remember, the radius is always a positive value.
• The length of 2 in our example tells us that every point is 2 units from the center \((0, 2)\).

This concept helps us accurately draw and understand the circle's reach on the plane.

Standard Form of a Circle Equation

The standard form of a circle's equation is a fundamental way of expression that allows simple identification of the circle's key components: its center and its radius.The formula is \[ (x - h)^2 + (y - k)^2 = r^2 \]and it is structured to provide both the center \((h, k)\) and the squared radius \(r^2\) at a glance.
To interpret this:

• The expression \((x - h)^2\) suggests a shift of \(h\) units along the x-axis.
• The expression \((y - k)^2\) indicates a shift of \(k\) units along the y-axis.
• \(r^2\) is the square of the radius.

For instance, the equation \[ x^2 + (y - 2)^2 = 4 \]matches this standard form, making it possible to extract the circle's center \((0, 2)\) and radius \(r = 2\).
This form is both a handy way to write circle equations and a tool for effortlessly determining the key properties of the circle.