Problem 58 - Algebra 2

Question

Find the center and the radius of each circle. $$ x^{2}+(y-4)^{2}=11 $$

Step-by-Step Solution

Answer

The center of the circle is at the point \((0,4)\) and the radius is \(\sqrt{11}\).

Step 1: Identify the Center

For the equation of a circle \((x-h)^{2}+(y-k)^{2}=r^{2}\), the center is given by the coordinates \((h, k)\). In the exercise equation \(x^{2}+(y-4)^{2}=11\), we have \(h=0\) and \(k=4\). Therefore, the center of the circle is at the point \((0,4)\).

Step 2: Find the Radius

In the equation of a circle, the radius is given by \(r\), which is the square root of the value on the right side of the equation. So, for the exercise equation \(x^{2}+(y-4)^{2}=11\), the radius \(r\) is \(\sqrt{11}\).

Key Concepts

Center of a Circle

To determine the center of a circle when you are given its equation, there's a helpful formula. The standard form of the equation of a circle is given by:

• \((x-h)^{2} + (y-k)^{2} = r^{2}\)

Here, the center of the circle is located at the point

• \((h, k)\)

In simpler terms, the values of \(h\) and \(k\) are taken from the parts of the equation where \((x - h)\) and \((y - k)\) appear. These are your coordinates for the center of the circle.
In the exercise equation \(x^{2}+(y-4)^{2}=11\), notice there is no \((x-h)\) term, so \(h\) is 0. The term \((y-4)\) reveals that \(k = 4\). Thus, the center is

• \((0, 4)\).

This method lets you find where the center of the circle lies swiftly.

Radius

The radius of a circle is the distance from its center out to any point on its boundary. Using the circle equation in standard form, we calculate this as the square root of the number on the right side of the equals sign.

• Equation: \((x-h)^{2}+(y-k)^{2} = r^{2}\)
• Radius: \(r\) is the square root of the constant on the right.

For example, in the equation \(x^{2}+(y-4)^{2}=11\), identify \(11\) as the value you need to take the square root of.
So, calculate:

• \(r = \sqrt{11}\)

This gives us the radius. Remember, radius is always a positive value as it represents physical distance.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, combines algebra and geometry using a coordinate system. This method allows us to solve geometry problems algebraically and understand geometric figures with the aid of coordinates.

• In the realm of circles, equations can represent a circle on a coordinate plane.
• Every point on a circle is equidistant from its center, a basis of this geometry.

By relating algebraic equations to coordinate geometry, you can explore circles by their equations as explored in this exercise. In our specific problem with the equation \(x^{2}+(y-4)^{2}=11\), the circle's representation involves breaking down the squared terms to find particular features like the center and radius.
Coordinate geometry empowers us to visualize, analyze, and interpret geometric shapes with precision.