Problem 22 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((-2,-1) ;\) radius \(=\sqrt{5}\)

Step-by-Step Solution

Answer

The equation of the circle with center \((-2, -1)\) and radius \(\sqrt{5}\) is: \[(x+2)^2 + (y+1)^2 = 5.\]

Step 1: Identify the center and radius of the circle

We are given that the center of the circle is \((-2, -1)\), which means \(a=-2\) and \(b=-1\). The radius of the circle is given as \(\sqrt{5}\), so \(r = \sqrt{5}\).

Step 2: Write the general equation of a circle

The standard equation of a circle is given by the formula \((x-a)^2 + (y-b)^2 = r^2\).

Step 3: Substitute the values of a, b, and r into the equation

Plugging in the values of \(a\), \(b\), and \(r\) into the standard equation of a circle, we get: \[(x-(-2))^2 + (y-(-1))^2 = (\sqrt{5})^2.\]

Step 4: Simplify the equation

Simplify the equation by expanding the terms and squaring the radius: \[(x+2)^2 + (y+1)^2 = 5.\]

Step 5: Write the final equation of the circle

The equation of the circle with center \((-2, -1)\) and radius \(\sqrt{5}\) is: \[(x+2)^2 + (y+1)^2 = 5.\]

Key Concepts

Center of a Circle

The center of a circle is a point that is equidistant from all points along the circle's edge. In an equation of a circle, the center is represented by a coordinate point \(a, b\). This point is crucial as it defines the exact position of the circle on the coordinate plane.

For example, if the center of a circle is \(-2, -1\), it provides the "center point" from which to measure the radius in all directions around the circle.

Determining the center is the first step in writing the equation of a circle because it tells us where the circle is located. Always remember: \(a\) and \(b\) are just the \(x\) and \(y\) coordinates of the center, not slopes or distances. The notation is key to inserting the correct values into the standard form equation, as these will affect the signs used in the calculations.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. It is one of the most fundamental properties of a circle and is always consistent throughout. In the equation of a circle, this radius is denoted by the variable \(r\).

When calculating and using the radius in circle equations, it might appear directly, as \( \), or indirectly, as \(r^2\), which is the square of the radius.

For instance, if the radius is given as \( = \sqrt{5}\), you remember that the squaring of this radius when substituted into the equation will convert \( \) into 5 because \((\sqrt{5})^2 = 5\). Therefore, identifying and correctly substituting the radius is key to deriving the correct form of the circle’s equation.

Standard Form of a Circle Equation

The standard form of a circle's equation is a simple and powerful representation in mathematics. This form makes it easy to clearly see the circle's properties, including its center and radius. The standard form is written as: \((x-a)^2 + (y-b)^2 = r^2\), where:

• \(a\) represents the \(x\)-coordinate of the center.
• \(b\) represents the \(y\)-coordinate of the center.
• \(r\) is the radius.

This arrangement ensures that any changes in center or radius are directly visible in the actual equation. They help in modeling circles in graphs and solving geometry problems efficiently.

Taking the given center \(-2, -1\) and radius \( = \sqrt{5}\), the standard form equation transforms into \((x+2)^2 + (y+1)^2 = 5\). This expression perfectly represents the circle on a Cartesian plane and provides all needed information to accurately describe and graph the circle.