Problem 20 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((4,-6) ;\) radius \(=3\)

Step-by-Step Solution

Answer

The equation of the circle with center \( (4, -6) \) and radius 3 is: \((x-4)^2 + (y+6)^2 = 9\).

Step 1: We are given the center of the circle as (4, -6) and the radius as 3. #Step 2: Use the general equation of a circle#

Recall that the general form of a circle's equation is: \((x-a)^2 + (y-b)^2 = r^2\) #Step 3: Substitute the center and radius values#

Step 2: Substitute a = 4, b = -6, and r = 3 into the equation: \((x-4)^2 + (y+6)^2 = 3^2\) #Step 4: Simplify the equation#

We can write the equation as: \((x-4)^2 + (y+6)^2 = 9\) The equation of the circle with center (4, -6) and radius 3 is: \((x-4)^2 + (y+6)^2 = 9\).

Key Concepts

Circle Center

The circle center is a crucial element when forming the equation of a circle. It defines the exact middle point from which every direction on the circle's boundary is equidistant. In the context of a two-dimensional plane, the center of the circle is represented as a point with coordinates in the form \( (a, b) \).

In our exercise, the center of the circle was provided as \( (4, -6) \). This means the circle is centered at 4 on the x-axis and -6 on the y-axis. Understanding this point helps in plotting the circle correctly on a graph.

• \( a = 4 \) - This is the x-coordinate of the center.
• \( b = -6 \) - This is the y-coordinate of the center.

The center affects how the circle is positioned within the coordinate grid. It's always important to correctly identify the circle's center to ensure the accuracy of any calculations or graphical representations.

Circle Radius

The radius of a circle is the distance from its center to any point on its boundary. It is a fundamental attribute because it determines the size of the circle. In the standard equation of a circle, the radius is denoted as \( r \).

In our example, the radius given is 3. This simply means that every point along the circle's perimeter is precisely 3 units away from the center \( (4, -6) \).

• It is always a positive value.
• It helps in calculating other geometrical properties like the diameter, which is twice the radius.

The knowledge of the radius aids in constructing the circle accurately, and in calculating its circumference and area for further assessments. Remember that the radius also plays a key role in forming the circle's area with the formula \( \pi r^2 \). It is essential for understanding the spatial spread of the circle.

Standard Form of a Circle

The standard form of the equation of a circle is an essential way to represent a circle algebraically. It allows you to see both the center and the radius at a glance, as they are embedded within the formula \( (x-a)^2 + (y-b)^2 = r^2 \).

This equation breaks down into meaningful parts:

• \( (x-a) \) and \( (y-b) \) indicate how the formula accounts for shifts from the origin to the circle's center \( (a, b) \).
• \( r^2 \) denotes the square of the radius, ensuring all points \( (x, y) \) maintain the constant distance from the center.

In our exercise, the standard form of the circle with center \( (4, -6) \) and radius 3 is shown as \( (x-4)^2 + (y+6)^2 = 9 \). Observe here:
The transformations:

• The minus sign in \( y-b \) turns into a plus sign for negative b-coordinates such as -6.
• The radius, upon squaring, becomes 9 (as \( 3^2 = 9 \)).

The standard form gives direct insight into the circle's properties, making it a powerful tool for graphing, solving equations, and enhancing your understanding of geometric relations.