Problem 14 - Algebra for College Students

Question

For Problems \(15-32\), find the center and the length of a radius of each of the circles. $$ (x-5)^{2}+(y-7)^{2}=25 $$

Step-by-Step Solution

Answer

Center: (5, 7); Radius: 5.

Step 1: Identify Circle Equation Form

The given equation is \((x-5)^2 + (y-7)^2 = 25\). This is in the standard form of a circle's equation, \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Determine the Circle's Center

By comparing \((x-5)^2 + (y-7)^2 = 25\) to the standard form \((x-h)^2 + (y-k)^2 = r^2\), we identify \(h = 5\) and \(k = 7\). Therefore, the center of the circle is \((5, 7)\).

Step 3: Calculate the Circle's Radius

The equation \((x-5)^2 + (y-7)^2 = 25\) gives us \(r^2 = 25\). To find \(r\), we take the square root: \(r = \sqrt{25} = 5\).

Step 4: Final Confirmation

The center of the circle is \((5, 7)\) and the radius is 5. Thus, this matches the identified components from the standard form of the circle's equation.

Key Concepts

Center of a Circle

Understanding the center of a circle is crucial in geometry when dealing with circle equations. The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\) Here, \((h, k)\) represents the center of the circle.

How to Find the Center

• Compare the given circle equation with the standard form.
• Identify the values h and k from \((x-h)^2\) and \((y-k)^2\).

In the exercise example, the equation is \((x-5)^2 + (y-7)^2 = 25\). By comparison:

• \(h = 5\)
• \(k = 7\)

Thus, the center of this circle is located at the point \((5, 7)\). Remember that the center is always given as an ordered pair \((h, k)\). Knowing the center helps us understand the positioning of the circle in the coordinate plane.

Radius of a Circle

The radius of a circle is the distance from its center to any point along its edge. In the standard equation of a circle, \((x-h)^2 + (y-k)^2 = r^2\), \(r\) is the radius. To find the radius, you take the square root of \(r^2\).

Calculating the Radius

• Identify \(r^2\) from the equation.
• Take the square root of that value to find \(r\).

For instance, in our exercise, the equation is \((x-5)^2 + (y-7)^2 = 25\).

Steps

• Here, \(r^2 = 25\).
• Thus, \(r = \sqrt{25} = 5\).

So, the radius of this circle is 5 units. Understanding the radius helps in various applications, such as calculating the circumference and area of the circle.

Standard Form of a Circle's Equation

The standard form of a circle's equation provides a neat and easy way to describe a circle on the Cartesian plane. The form is: \((x-h)^2 + (y-k)^2 = r^2\). Here are its key components:

• \((h, k)\): The center of the circle.
• \(r\): The radius of the circle.

This form allows us to quickly identify both these characteristics.

Using the Standard Form

When you compare a circle's equation to this standard form, you can immediately determine its center and radius. Like in our given problem:

• Equation: \((x-5)^2 + (y-7)^2 = 25\).
• Compare and identify: \(h = 5\), \(k = 7\), \(r^2 = 25\).
• The center is \((5, 7)\) and radius is 5.

Using the standard form, you can also graph the circle easily, locate its center on a coordinate plane, and understand its dimensions. Converting other forms of circle equations (like general form) into this standard form helps simplify these processes.