Problem 24 - The Calculus with Analytic Geometry

Question

Find an equation whose graph is the circle that is the set of all points that are at a distance of 4 units from the point \((1,3)\)

Step-by-Step Solution

Answer

(x - 1)^2 + (y - 3)^2 = 16

Step 1: Identify Center and Radius

The center of the circle is given by the point \( (1, 3) \) and the radius is 4 units.

Step 2: Recall the Equation of a Circle

The equation of a circle with center at \( (h, k) \) and radius \( r \) is \[ (x - h)^2 + (y - k)^2 = r^2. \]

Step 3: Substitute Center and Radius

Substitute \( h = 1 \), \( k = 3 \), and \( r = 4 \) into the equation: \[ (x - 1)^2 + (y - 3)^2 = 4^2. \]

Step 4: Simplify the Equation

Simplify the radius part of the equation: \[ (x - 1)^2 + (y - 3)^2 = 16. \]

Key Concepts

Center

Understanding the center of a circle is crucial when working with its equation. The center of a circle is simply a point in the two-dimensional plane. It is usually denoted as \( (h, k) \).

In our example, we are given the center of the circle as \( (1, 3) \). This point represents where the circle is perfectly balanced in the middle of the plane.

If you were to draw the circle, you would place your compass point on this exact coordinate and stretch out to the radius to draw the circle. The center (h, k) essentially determines where the circle is positioned on the graph.

Radius

The radius is another key concept when dealing with circles. The radius is the distance from the center of the circle to any point on the circle. In simpler terms, it’s half the width of the circle when measured through the center.

For our problem, the radius has been given as 4 units. This means that if you start at the center, \( (1, 3) \), and move 4 units in any direction, you will land on the circle.

• Every point on the circle is exactly 4 units away from \( (1, 3) \).

Being aware of the circle's radius helps in graphing it accurately and understanding its size.

Standard Form of a Circle

One of the most important formulas to remember in dealing with circles is their standard form equation. The standard form of a circle’s equation is:
\[ (x - h)^2 + (y - k)^2 = r^2. \]
Here’s what each part represents:

• \( h \) and \( k \) are the x and y coordinates of the circle’s center
• \( r \) is the radius of the circle

For our example, we substitute the center \( (1, 3) \) and the radius 4 into the standard form equation:
\[ (x - 1)^2 + (y - 3)^2 = 4^2. \]
Simplifying, we have:
\[ (x - 1)^2 + (y - 3)^2 = 16. \]
This is the standard form equation for our circle. Understanding the standard form helps us quickly identify the circle’s center and radius, as well as graph it accurately.