Problem 27 - Intermediate Algebra

Question

Write the equation of a circle in standard form with the following properties. Center at \((-2,6) ;\) radius 12

Step-by-Step Solution

Answer

The equation of the circle is \((x + 2)^2 + (y - 6)^2 = 144\).

Step 1: Identify Standard Form of Circle Equation

The standard form of the equation of a circle is given by \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Substitute Center into the Equation

Substitute the center of the circle \((-2, 6)\) into the standard form equation as \(h = -2\) and \(k = 6\), so the equation becomes \((x + 2)^2 + (y - 6)^2 = r^2\).

Step 3: Substitute Radius into the Equation

Substitute the radius \(r = 12\) into the equation. Since \(r^2 = 12^2 = 144\), the equation becomes \((x + 2)^2 + (y - 6)^2 = 144\).

Key Concepts

Standard Form

The standard form of a circle's equation is a key element in understanding how circles are represented mathematically. This form is expressed as \((x - h)^2 + (y - k)^2 = r^2\). It breaks down into two primary parts: the center of the circle, represented by \((h, k)\), and the radius, represented by \(r\).
This form is particularly useful because it describes every point \((x, y)\) on the circle's perimeter, establishing a precise relationship between the circle's center and radius.
By utilizing this equation, you can easily recognize or calculate the specific components needed for circle-related problems, such as the one above, where substituting values gives you the full equation describing the circle's position and size.

Circle Center

The circle's center is pivotal in its geometric representation. In a standard form equation, the center is denoted by the coordinates \((h, k)\).
This point acts as the reference location from which all points on the circle are equally distant.
Understanding the center's role helps in aligning the circle correctly on a coordinate plane, making it easier to visualize and solve related mathematical problems.
To determine the specific center of a circle with a given equation, you simply extract the \(h\) and \(k\) values, which have already been adjusted to account for the subtraction within the formula.

• For an equation like \((x - h)^2 + (y - k)^2 = r^2\), the center is \((h, k)\).
• In our example that was altered to \((x + 2)^2 + (y - 6)^2 = r^2\), the center is therefore \((-2, 6)\).

Circle Radius

The radius is an essential feature that defines how large or small a circle is. In the standard form of a circle's equation, \(r\) represents this radius.
This value is squared in the equation, making the formula appear as \(r^2\).
Understanding the radius involves comprehending its role as the distance from the circle's center to any point on its edge.

• To find the radius from the equation, take the square root of the \(r^2\) value. For example, if given \(r^2 = 144\), the radius \(r\) is 12.
• Knowing the radius, we can easily calculate the circle's circumference and area, should we need to explore further geometrical properties.

With these concepts, you are equipped to write or interpret any circle equation in standard form efficiently.