Problem 27 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((0,-4) ;\) radius \(=2 \sqrt{2}\)

Step-by-Step Solution

Answer

The equation of the circle with center \((0,-4)\) and radius \(2\sqrt{2}\) is \(x^2 + (y + 4)^2 = 8\).

Step 1: Plug values into the general equation

Let's plug the given values in the general equation: \[(x - 0)^2 + (y + 4)^2 = (2 \sqrt{2})^2\] Since the center is at \((0, -4)\) and the radius is \(2\sqrt{2}\).

Step 2: Simplify the equation

Now, let's simplify the expression: \[x^2 + (y + 4)^2 = 8\] This is the equation of the circle with the given center \((0,-4)\) and radius \(2\sqrt{2}\).

Key Concepts

Understanding Circle Center

The center of a circle is a crucial point that determines its position in a coordinate plane. It is defined as the midpoint from which every point on the circle is equidistant. In this exercise, the center is at the coordinates \((0, -4)\). This means the circle is located 4 units below the origin along the y-axis. Knowing the circle center allows us to write the equation of the circle easily. It's usually represented in the equation format as \((h, k)\), where \(h\) and \(k\) are the x and y coordinates, respectively.

Calculating the Radius

The radius of a circle is the distance from the center to any point on the circle. For this circle, the radius is given as \(2\sqrt{2}\). This value is vital as it helps define the size of the circle. The radius is squared when used in the circle equation, which in this case becomes \((2\sqrt{2})^2 = 8\). Understanding how to work with the square root values and square them is an essential skill in algebra. It ensures you correctly input these values into the equation.

Relevance of Algebra

Algebra is the mathematics of symbols and rules for manipulating those symbols. In the context of circles, algebra helps us form the equation of the circle. By using algebraic expressions, we represent geometric ideas such as distance and area. Here, the equation \(x^2 + (y+4)^2 = 8\) is derived using algebraic techniques. It simplifies the understanding and representation of the circle, letting us predict any point lying on its circumference.

Role of Coordinate Geometry

Coordinate geometry, or analytic geometry, combines geometry and algebra by using a coordinate system. It's a powerful tool for solving geometric problems with algebraic equations. In circles, coordinate geometry allows us to position the circle in the plane and describe its properties precisely. The equation \((x-h)^2 + (y-k)^2 = r^2\) ensures that any point \((x, y)\) maintaining this equality actually lies on the circle, with the center \((h, k)\) and radius \(r\). By understanding this equation, we can easily graph circles and determine their characteristics just from the algebraic expressions.