Problem 55 - Calculus for Biology and Medicine

Question

Find the equation of a circle with center \((1,-2)\) and radius 2 .

Step-by-Step Solution

Answer

The equation of the circle is \((x - 1)^2 + (y + 2)^2 = 4\).

Step 1: Recall the Standard Form of a Circle's Equation

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) represents the coordinates of the center, and \(r\) is the radius of the circle.

Step 2: Identify Given Values

Given the center of the circle \((h,k) = (1,-2)\) and the radius \(r = 2\), you need to substitute these values into the standard form equation.

Step 3: Substitute Values into the Equation

By substituting the center \((1, -2)\) and the radius 2 into the equation \((x - h)^2 + (y - k)^2 = r^2\), we get \((x - 1)^2 + (y + 2)^2 = 2^2\).

Step 4: Simplify the Equation

Square the radius \(r = 2\) to find \(r^2 = 4\). Thus, the equation becomes \((x - 1)^2 + (y + 2)^2 = 4\).

Key Concepts

standard form of a circle

The standard form of a circle is a neat way to express the equation of a circle in math. It's expressed as \((x - h)^2 + (y - k)^2 = r^2\). This formula helps us quickly identify key features of the circle. Here are the parts:

• \((x - h)\)^2: The horizontal part of the formula. It shows how much the circle is moved from the y-axis.
• \((y - k)\)^2: This is the vertical component. It indicates the circle's shift from the x-axis.
• \(r^2\): It's the square of the radius. This represents the size of the circle.

When given a circle's equation in this form, you can easily find the center and the radius, which are vital to understanding a circle's properties.

coordinates of the center

The coordinates of the center of a circle, represented as \((h, k)\), define where the circle is located on the Cartesian plane. This is like the heart of the circle, from which all points are equidistant by the radius length. In the equation \((x - h)^2 + (y - k)^2 = r^2\), the

• \(h\) is the x-coordinate of the center.
• \(k\) is the y-coordinate of the center.

If the equation is \((x - 1)^2 + (y + 2)^2 = 4\), the center is at \((1, -2)\). This means the circle is positioned one unit to the right of the y-axis and two units below the x-axis.

• Moving right and left along the x-axis affects the \(h\).
• Moving up and down along the y-axis affects the \(k\).

Understanding these coordinates is essential for graphing and visualizing where your circle sits on the graph.

radius of a circle

The radius of a circle is a crucial measurement that tells us about the circle's size. It's the distance from the circle's center to any point on the circle. In our equation form \((x - h)^2 + (y - k)^2 = r^2\), the radius is represented by \(r\).For example, if \(r^2 = 4\), we find the radius by taking the square root: \(r = \sqrt{4} = 2\). This shows us that every point on the circle is exactly 2 units away from the center, creating a perfect round shape.

• Radius is always a positive number and represents length.
• A larger radius means a larger circle.
• All points on the circle are equidistant from the center.

This concept is vital since the radius influences the entire shape and size of the circle, making it a foundational concept in geometry.