Problem 20 - Algebra 2

Question

For each equation, find the center and radius of the circle. $$ (x+2)^{2}+(y-10)^{2}=4 $$

Step-by-Step Solution

Answer

Center of circle is at (2, 10) and radius is 2.

Step 1: Determine Center

By simply looking at the equation, it can be seen that the center coordinates (h, k) are given by the values within the parentheses. So, h is opposite in sign to the value with x and k is opposite in sign to the value with y. Here, h is -(-2) = 2 and k is -(-10) = 10. So, the center of the circle is (2, 10).

Step 2: Determine Radius

The radius r is simply the square root of the number on the other side of the equation. Here, r is \(\sqrt{4}\) = 2.

Key Concepts

Center of a Circle

The center of a circle is a crucial point as it defines the position around which the circle is perfectly symmetric. If your circle equation is in the standard form, identifying the center becomes straightforward. By standard form, we mean:

• \((x - h)^2 + (y - k)^2 = r^2\)

The coordinates \((h, k)\) represent the center of the circle. It's important to note that these values are not simply the numbers inside the parentheses; they are the opposite of these numbers. By examining the signs, you can accurately locate the center:

• If the equation is \((x - 2)^2 + (y - 10)^2\), then the center \((h, k)\) is actually \((2, 10)\).

Think of it as reversing the sign for both x and y to find the center coordinates. This is because the form of the equation compels a subtraction. If the equation shows addition, it indicates that the original term was negative.

Radius of a Circle

The radius is the distance from the circle's center to any point on its boundary. It is a length, so it cannot be negative. In the equation, after the equality sign, the number you see represents the radius squared, or \(r^2\).

• For the equation \((x+2)^{2}+(y-10)^{2}=4\), this tells us that \(r^2 = 4\).

To determine the radius, simply find the square root of this number. Calculating the square root of 4 gives us:

• \(r = \sqrt{4} = 2\)

Remember, since the radius is a measure of length, it's always positive. Use this method to easily extract the radius from any standard circle equation.

Standard Form of a Circle Equation

The standard form of a circle equation is an intuitive way of writing a circle's equation as it directly provides useful information about the circle. The equation is structured as follows:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here, \(h, k\) are the coordinates of the circle's center, and \(r\) is the radius.

• This form clearly separates the circle's characteristics by incorporating them into a straight-forward algebraic statement.
• By presenting the center's coordinates and the square of the radius explicitly, you can quickly identify a circle's position and size.

This form helps simplify problem-solving as it transforms into a recognizable pattern. Recognizing this pattern will make finding the center and radius much easier, saving you time and effort in the long run.