Problem 46 - Algebra 2

Question

Use the given information to write an equation of the circle. \(\operatorname{area} 25 \pi,\) center \((5,-3)\)

Step-by-Step Solution

Answer

The equation of the given circle is \((x-5)^{2}+(y+3)^{2}=25\).

Step 1: Calculate the radius of the circle

The provided area of the circle is \(25 \pi\). The formula of the area of a circle is \(\pi r^{2}\), where r is the radius. Set these two equal to solve for the radius, which gives \(r^{2} = 25\), so \(r = 5\).

Step 2: Write the equation of the circle using the center and radius

Now, substitute the center point (5, -3) and the radius (5) into the standard form of the circle equation, \((x-a)^{2}+(y-b)^{2}=r^{2}\). This gives the equation \((x-5)^{2}+(y+3)^{2}= 25\).

Key Concepts

Circle Area

The area of a circle is one of the most fundamental properties used to understand and describe circles. It is defined as the amount of space enclosed within the boundaries of the circle. The formula to calculate the area of a circle is \[ A = \pi r^2 \]where

• \( A \) represents the area of the circle,
• \( \pi \) is a mathematical constant approximately equal to 3.14159, and
• \( r \) is the radius of the circle.

For example, if the area of the circle is given as \(25\pi\), it implies that the area enclosed by the circle is exactly 25 times the constant \(\pi\). The area formula is derived from the relationship between the circle's diameter, its circumference, and ultimately, the radius. Understanding this formula is crucial because it directly links to how we calculate the size of a circle in real-world applications like finding the surface area of a round table or determining the space within a circular garden.

Radius Calculation

Finding the radius of a circle, when its area is given, is straightforward using the area formula. With the given area, we utilize the relationship:\[ A = \pi r^2 \]Rearranging the formula helps us solve for the radius:\[ r^2 = \frac{A}{\pi} \]Since the area is \(25\pi\), substitute this into the equation:\[ r^2 = \frac{25\pi}{\pi} = 25 \]Thus, we find

• \( r^2 \) equals 25,
• so \( r \) is the square root of 25,
• resulting in \( r = 5 \).

Here, we've learned that the radius of the circle is 5. This value not only tells us the distance from the center of the circle to any point on its edge, but it also serves as a vital parameter for constructing the circle's equation in standard form. Getting the radius right is critical for drawing accurate geometrical representations and solving problems related to circle geometry.

Standard Form of Circle Equation

In geometry, many circles can be expressed in what's known as the standard form of a circle equation. This equation provides a neat representation of the circle's properties, specifically its location and size. The standard form is given by:\[ (x-a)^2 + (y-b)^2 = r^2 \]Where

• \((a, b)\) is the center of the circle,
• \(x\) and \(y\) are variables representing any point on the circle, and
• \(r\) is the radius.

For the example where the center is \((5, -3)\) and the radius is 5, we substitute these values into the equation:\[ (x-5)^2 + (y+3)^2 = 25 \]This format makes it easy to identify the circle's center and radius, which are crucial for graphing the circle or performing further calculations. The standard form is especially useful when analyzing circles in the context of the Cartesian coordinate system, as it aligns neatly with the grid, facilitating easier visualization and measurement.