Problem 33 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$\text { Center }(3,2), r=5$$

Step-by-Step Solution

Answer

The standard form of the equation of the circle with center (3,2) and radius 5 is \((x-3)^2 + (y-2)^2 = 25\).

Step 1: Identify the values of h, k and r

In the given exercise, the center of the circle is represented by (3,2). Therefore, \(h=3\) and \(k=2\). The radius of the circle \(r=5\).

Step 2: Substitute h, k and r values into the standard form of the equation of a circle

Substitute \(h=3\), \(k=2\) and \(r=5\) into the standard equation of a circle \((x-h)^2 + (y-k)^2 = r^2\), the resulting equation is: \((x-3)^2 + (y-2)^2 = 5^2\).

Step 3: Simplify the equation

Simplify \(5^2\) to get 25 to form the final equation of the circle \((x-3)^2 + (y-2)^2 = 25\) which is the standard form of the equation of the circle with center (3,2) and radius 5.

Key Concepts

Understanding the Standard Form of a Circle's Equation

To express a circle's equation in standard form is like giving it a proper "address". This helps us easily identify where the circle is centered and how big it is. The standard form formula is:

• \((x-h)^2 + (y-k)^2 = r^2\)

Here’s what each part represents:

• \(h\) and \(k\) are the coordinates of the circle's center \((h, k)\).
• \(r\) is the radius of the circle.

Substitute the given center and radius values into the formula. By doing this, you convert the circle's characteristics into an equation that clearly describes its position and size on a coordinate plane. Recognizing this format helps in many geometric calculations and analyses.

Finding the Center and Radius of a Circle

In any circle equation, identifying the center and radius is crucial. This step helps set the parameters for the standard form equation. Let’s break down this exercise:

• The center of the circle is given as \( (3, 2) \), so \(h = 3\) and \(k = 2\).
• The radius is provided as \(r = 5\).

Once you have these values, you can easily plug them into the equation to form the circle's equation in standard form. Knowing the center tells you where the circle is located on the XY-plane, and the radius tells you its size. Together, they define the circle's unique position and dimensions.

Simplifying Equations for Clarity

Simplifying the circle's equation is an important step for creating a clear and understandable equation. Let's consider the given exercise:

• The standard form starts with \((x-3)^2 + (y-2)^2 = 5^2\).
• Multiplying \(5^2\) results in \(25\).

This simplification leads to the well-organized equation \((x-3)^2 + (y-2)^2 = 25\). A simplified form is not just about aesthetics; it makes the equation more practical and easier to use. Having the simplified number for the radius squared shows exactly how a circle relates to the rest of the coordinate plane, helping solve further problems and understanding the circle's properties straightforwardly.