Question

Find the standard form of the equation of each circle. Center (2,-4) and circumference \(16 \pi\)

Short answer

(x-2)^2 + (y+4)^2 = 64

Step-by-step solution

Determine the radius from the circumference

The formula to find the circumference of a circle is given by \text{Circumference} = 2\pi r. We know the circumference is 16\pi. Set up the equation as follows: \[ 2 \pi r = 16 \pi \]Solve for r by dividing both sides by 2\pi: \[ r = \frac{16\pi}{2\pi} = 8 \]So, the radius of the circle is 8.

Use the standard form of a circle's equation

The standard form of the equation of a circle is \[ (x-h)^2 + (y-k)^2 = r^2 \]where (h, k) are the coordinates of the center, and r is the radius. From the problem, we have the center (2, -4) and radius 8.

Write the equation of the circle

Substitute the center (2, -4) and the radius 8 into the standard form equation: \[ (x-2)^2 + (y+4)^2 = 8^2 \]Simplify the equation: \[ (x-2)^2 + (y+4)^2 = 64 \]Thus, the standard form of the equation of the circle is \[ (x-2)^2 + (y+4)^2 = 64 \]

Key concepts

standard form

The standard form of a circle's equation is essential to identifying and working with circles in algebra. It is written as \((x-h)^2 + (y-k)^2 = r^2\).
In this equation:
- \(h\) and \(k\) are the coordinates of the circle's center.
- \(r\) is the radius of the circle.
This form helps to easily locate the center and radius of the circle, which are crucial for solving problems or graphing the circle.
For example, if we have a circle with center \((2, -4)\) and a radius of 8, the equation would be \((x-2)^2 + (y+4)^2 = 64\).
This means every point \(x, y\) that satisfies this equation lies on the circle.

circle radius

Understanding the radius of a circle is fundamental in solving circle-related problems.
The radius \(r\) is the distance from the center of the circle to any point on the circle's edge.
It plays a vital role in various equations related to circles.
For instance, if you're given the circumference of a circle, you can find the radius using the formula \(\text{Circumference} = 2\pi r\).
Look at our example where the circumference is \(16\pi\).
You can set up the equation: \2 \pi r = 16 \pi\.
Solving this equation, \r = 8\.
Knowing the radius allows us to write the circle's standard equation and to understand any geometrical properties connected to the circle.

circumference formula

The circumference formula is critical for finding the distance around a circle.
It is given by \(\text{Circumference} = 2 \pi r\).
This formula connects the radius directly to the circumference.
To solve for the radius when you know the circumference, rearrange the formula: \ r = \frac{\text{Circumference}}{2 \pi} \.
In our case, for a circle with a circumference of \(16 \pi\), the calculation would be \ r = \frac{16 \pi}{2 \pi} = 8 \.
Understanding this formula helps in transitioning between the circumference and the radius, which is useful in various applications, from simple geometry problems to real-world contexts like designing circular objects.