Problem 35 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$\text { Center }(-1,4), r=2$$

Step-by-Step Solution

Answer

So, the standard form of the equation of the circle with center (-1,4) and radius 2 is \( (x + 1)^2 + (y - 4)^2 = 4\).

Step 1: Identify the given values

The exercise provides these values: The center of the circle (-1,4) and radius 2. Therefore, in our case a=-1, b=4 and r=2.

Step 2: Substitute the given values into the standard form

Now we can substitute the given values into the standard form \( (x-a)^2 + (y-b)^2 = r^2 \). This gives us: \( (x - (-1))^2 + (y - 4)^2 = 2^2 \)

Step 3: Simplify the equation

Simplify the equation to its most basic form: \( (x + 1)^2 + (y - 4)^2 = 4\).

Key Concepts

Understanding the Standard Form of a Circle Equation

In geometry, the standard form of a circle equation helps describe the circle's position and size using its center and radius. The formula is essential for determining various properties of a circle. It's given by:\[(x-a)^2 + (y-b)^2 = r^2\]Here's what each term represents:

• \(x\) and \(y\) are variables corresponding to the coordinates of any point on the circle.
• \(a\) and \(b\) are the x and y coordinates of the center of the circle.
• \(r\) represents the radius of the circle.

To write the equation of a circle, you substitute the values for \(a\), \(b\), and \(r\) into this formula, as shown in the solution: \((x + 1)^2 + (y - 4)^2 = 4\). This form helps identify the circle's precise location in a coordinate plane. It's crucial for understanding how circles are geometrically modeled.

Locating the Center of a Circle

The center of a circle is a fundamental component in its equation and determines where it's positioned within a coordinate plane. In the standard form \((x-a)^2 + (y-b)^2 = r^2\), the values \(a\) and \(b\) are the x and y coordinates of the center, commonly denoted as \((a, b)\).

For instance, if the center is given as \((-1,4)\), it indicates:

• The circle is centered at \(x = -1\) on the x-axis.
• The circle is centered at \(y = 4\) on the y-axis.

This central point is pivotal as it serves as a reference point from which the radius extends in all directions. Locating the center allows us to visualize where the circle starts, aiding in graphing and analytical reasoning involving circles.

Determining the Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. In the standard form of a circle's equation, \((x-a)^2 + (y-b)^2 = r^2\), the term \(r^2\) represents the square of the circle's radius.

To determine the radius, you simply need to take the square root of \(r^2\). For the given example, the radius \(r\) is 2, which yields:

• \(r^2 = 2^2 = 4\).

The radius helps in defining the size of the circle and is crucial for constructing the circle accurately on a graph.
Understanding the radius also plays a pivotal role in calculating other circle properties, such as its diameter (which is twice the radius) and circumference (using the formula \(2\pi r\)). This extends our mathematical understanding and application of circles in various contexts.