Problem 89 - College Algebra and Calculus: An Applied Approach

Question

Find the standard form of the equation of the specified circle. Center: \((-1,2)\); point on circle: \((0,0)\)

Step-by-Step Solution

Answer

The standard form of the equation of the circle is \((x+1)^2 + (y-2)^2 = 5\).

Step 1: Calculate the radius

First, calculate the radius of the circle using the distance formula. Given the center of the circle \((-1,2)\) and a point on the circle \((0,0)\), the radius \(r\) can be computed as \(r= \sqrt{(0-(-1))^2 + (0-2)^2} = \sqrt{1 + 4} = \sqrt{5}\)

Step 2: Plug radius and center into circle equation

Substitute the values of the center and the radius into the standard equation of the circle to end up with \((x-(-1))^2 + (y-2)^2 = (\sqrt{5})^2\)

Step 3: Simplify the equation

Finally, simplify the equation to its standard form: \((x+1)^2 + (y-2)^2 = 5\)

Key Concepts

Circle Geometry

Understanding the geometry of a circle is fundamental in mathematics, especially when working with equations describing circles. A circle is a set of all points in a plane that are at a fixed distance, known as the radius, from a certain point, called the center. The standard form equation for a circle is \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h,k) \) is the center of the circle and \( r \) is the radius.

When given a problem like the one in our exercise, identifying the center and calculating the radius are the key steps before writing the equation. By knowing the distance formula to find the radius (which we'll discuss further below), and how to square the radius to complete the equation, we can fully describe the properties of that circle in the form of a neat, standard equation.

Radius of a Circle

The radius of a circle is not only the distance from the center to any point on the circle, but it's also a crucial measure that helps us in various aspects of circle geometry. For instance, when calculating the area \( A \) of a circle (using \( A=\pi r^2 \)) or its circumference \( C \) (using \( C=2\pi r \)), the radius \( r \) is an indispensable value.

In the context of our textbook exercise, the radius helped us articulate the standard form equation of the circle. It's important for students to understand that the radius can be any non-negative real number, and can be derived from just knowing the center of a circle and any other point on its perimeter by applying the distance formula.

Distance Formula

The distance formula is a tool based on the Pythagorean theorem that allows us to calculate the distance between any two points in a coordinate plane. It is represented by \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. It becomes especially useful in the context of circle equations.

For our exercise, the distance formula revealed the radius of the circle by calculating the distance between the center and a point on the circle. It's a straightforward method, yet powerful enough to unlock a necessary component for constructing the circle's standard form equation. Students should practice this formula to ensure they are comfortable with finding distances in a variety of geometric contexts.