Problem 32 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(0,0), r=8 $$

Step-by-Step Solution

Answer

The standard form of the equation of the circle with center \((0,0)\) and radius \(r=8\) is \(x^2 + y^2 = 64\).

Step 1: Identify the given information

The center of the circle is given as \((0,0)\) and the radius is given as \(r=8\). Therefore, we have \(h=0\), \(k=0\) and \(r=8\).

Step 2: Substitute the values into the equation

Substitute \(h\), \(k\) and \(r\) into the standard form of a circle equation \((x-h)^2 + (y-k)^2 = r^2\). This becomes \((x-0)^2 + (y-0)^2 = 8^2\). Simplifying further, we get \(x^2 + y^2 = 64\).

Step 3: Write final equation

The standard form of the equation for this circle is therefore \(x^2 + y^2 = 64\).

Key Concepts

Standard form

The standard form of the equation of a circle is a foundational concept in geometry. It is given by the formula \[ (x - h)^2 + (y - k)^2 = r^2 \] where

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

This form is very useful because it directly shows the center and the radius of the circle, making it a straightforward way to understand and graph circles on the Cartesian plane.
The equation results from the Pythagorean theorem applied in the setting of a circle.
Each point \((x, y)\) on the circle satisfies this equation, meaning every point is exactly \(r\) units away from the center \((h, k)\). This characteristic is what defines a circle, making the standard form equation a simple yet powerful tool in math.

Radius

The radius of a circle is a line segment that extends from the center to any point on the circle itself. It is a crucial measurement in circular geometry because it defines the size of the circle.
In the earlier exercise, you have learned how to use this information in the equation. The radius squared, \(r^2\), is used in the equation of the standard form of the circle, \[ (x - h)^2 + (y - k)^2 = r^2 \].

• Given a radius of 8, we compute \(r^2\) as 64 because \(8^2 = 64\).
• This value appears on the right side of the standard form equation.

Understanding how the radius connects to the equation helps visualize the circle and makes calculations, such as finding the area or circumference, straightforward.
The radius affects not only the size but helps in defining the circle's properties and behavior in different problems within algebra and geometry.

Coordinates of center

The coordinates of the center of a circle, denoted as \((h, k)\), are central to understanding its placement on the Cartesian plane. This point is where all lines radiating from the circle to its circumference meet.
In our exercise, the center is at \((0, 0)\), which simplifies our equation significantly.

• If you shift the center to a different point, it just changes the values of \(h\) and \(k\) in the formula \((x-h)^2 + (y-k)^2 = r^2\).
• The concept of the center coordinates allows you to predict and manipulate the circle's position.

Any change in these values shifts the circle horizontally or vertically but does not affect its shape or size.
The center is the reference point for defining the direction and extent of the radius and important for graphing the circle accurately.

Algebra

Algebra plays a significant role in solving and understanding the equation of a circle. It involves various operations like substitution, squaring numbers, and simplifying expressions.

• Substituting known values for the variables \((h, k, r)\) into the equation helps derive the specific standard form for a given circle.
• Simplifying expressions is essential, such as moving from \((x-0)^2 + (y-0)^2 = 8^2\) to \(x^2 + y^2 = 64\).This process makes the algebraic expression neat and easy to interpret.

Through algebraic manipulation, the equation becomes a usable tool for various applications, from graphing to solving complex problems involving multiple circles.
These algebraic skills are crucial for accurately representing the equation and leveraging it in broader mathematical contexts.