Problem 92 - College Algebra Essentials

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. The graph of \((x-4)+(y+6)=25\) is a circle with radius 5 centered at \((4,-6)\)

Step-by-Step Solution

Answer

The statement 'The graph of \( (x-4)^2 + (y+6)^2 = 25 \) is a circle with radius 5 centered at (4,-6)' makes sense because it matches the standard form of the equation of a circle and therefore correctly describes the properties of the circle.

Step 1: Understand the given problem

The given statement is \( (x-4)^2 + (y+6)^2 = 25 \). The claim is that this is a circle with radius 5, centered at (4,-6).

Step 2: Compare with standard form

The standard form for the equation of a circle is \( (x-a)^2 + (y-b)^2 = r^2 \), where the center of the circle is (a, b) and the radius is r. Comparing, we see that a=4, b=-6, and r^2 = 25.

Step 3: Evaluate the claim

We can see that r^2 = 25, therefore r = 5, which is the stated radius. Furthermore, the center of the circle is indeed (4,-6) as stated. Therefore, the claim made in the statement that the equation represents a circle of radius 5 centered at (4,-6) is correct.

Key Concepts

Standard Form

The standard form of an equation for a circle is very useful. It helps in identifying the precise location and size of the circle on a coordinate plane. The equation is written as \((x-a)^2 + (y-b)^2 = r^2\).

This form directly reveals:

• The center of the circle, which is at the coordinates \((a, b)\). This is derived from the terms \((x-a)^2\) and \((y-b)^2\).
• The radius of the circle, \(r\), which is the square root of the number on the right side of the equal sign, \(r^2\).

The equation must always start with squared terms on the left and a single number on the right. This helps in maintaining the structure and symmetry of a circle.

Radius

Understanding the radius \(r\) is crucial for interpreting a circle's equation accurately. The radius represents the distance from the center of the circle to any point on the edge of the circle.

In the standard form equation, \((x-a)^2 + (y-b)^2 = r^2\), you identify \(r\) by taking the square root of \(r^2\).
For example, if \(r^2 = 25\), then \(r = \sqrt{25} = 5\).

This value is useful not only in graphing but also in solving real-world applications like physics problems or engineering designs. The consistency of the radius gives the circle its perfectly round shape.

Center of a Circle

Locating the center of a circle from its equation is straightforward due to the clear structure of the standard form equation. The center is indicated by the coordinates \((a, b)\), found by examining the expressions \((x-a)^2\) and \((y-b)^2\).

In our example equation, \((x-4)^2 + (y+6)^2 = 25\), the center is at \((4, -6)\). The center acts as the reference point for all points on the circle.

Understanding the center's location helps in tasks like graphing the circle or modifying the equation when shifting positions on a plane. It is the first step in fully interpreting or creating any circle's equation.