Problem 45 - Algebra 2

Question

Use the given information to write an equation of the circle. radius \(7,\) center \((-6,13)\)

Step-by-Step Solution

Answer

The equation of the circle is \((x + 6)^2 + (y - 13)^2 = 49\).

Step 1: Identify the center and radius

First, identify the center \((-6,13)\) and the radius \(7\).

Step 2: Substitute the values into the circle equation

Next, substitute the values of \(h, k, r\) into the circle equation. So, \((x + 6)^2 + (y - 13)^2 = (7)^2\).

Step 3: Simplify the equation

Finally, simplify the square of the radius \((7)^2\) to get \(49\). Hence, the equation of the circle is \((x + 6)^2 + (y - 13)^2 = 49\).

Key Concepts

Radius of a Circle

The radius of a circle is a key component that defines its size. It is the distance from the center of the circle to any point on its circumference. This distance remains the same no matter which point on the circle you choose to measure from. In our exercise, the radius is given as 7.

Understanding the concept of radius is important because it helps in determining the area and circumference of the circle too. Also, a known radius is crucial when writing the equation of the circle in its standard form.

• The radius connects the center to the boundary of the circle.
• Algebraically, in the circle equation \((x-h)^2 + (y-k)^2 = r^2\), \(r\) stands for the radius.
• The square of the radius gives insight into the spatial extent of the circle.

Center of a Circle

The center of a circle is a unique point equidistant from all points on the circumference. In most mathematical contexts, the center gives the circle its unique position in the coordinate plane. For our exercise, the center is located at the point \((-6, 13)\).

Knowing the center allows us to describe the circle's location precisely. It also plays a critical role in formulating the equation of the circle.

• In the standard form equation \( (x-h)^2 + (y-k)^2 = r^2 \), \(h\) and \(k\) represent the x and y coordinates of the center.
• Shifting the center changes only the position of the circle, not the radius.
• The point is the pivot around which the circle expands, marked in the coordinate plane.

Standard Form of a Circle Equation

The standard form of a circle's equation is a way of writing the circle's properties using algebra. It neatly encapsulates details about the center and the radius. This form is given by \( (x-h)^2 + (y-k)^2 = r^2 \), where \(h\) and \(k\) are the x and y coordinates of the center, and \(r\) is the radius of the circle.

In our exercise, substituting \((-6, 13)\) for the center and 7 for the radius, we derived \( (x+6)^2 + (y-13)^2 = 49 \).

• This form helps in graphing the circle since it clearly shows the center.
• The equation indicates that squaring the radius provides the circle's area on a coordinate plane.
• Manipulating the equation makes transformations like translations and scalings easier.

Understanding this form is fundamental in various applications, from geometry to engineering, where circular elements are prevalent.