Problem 7 - Algebra 2

Question

Write an equation of a circle with the given center and radius. Check your answers. $$ (-3,0), 8 $$

Step-by-Step Solution

Answer

The equation of the circle with center (-3,0) and radius 8 is \((x+3)^2 + y^2 = 64\)

Step 1: Identify the given values

The problem gives the center of the circle as (-3,0) and the radius as 8. In the general equation for a circle, this corresponds to h = -3, k = 0, and r = 8.

Step 2: Substitute the values into the general equation

Substituting h = -3, k = 0, and r = 8 into the general equation \((x-h)^2 + (y-k)^2 = r^2\), we get \((x-(-3))^2 + (y-0)^2 = 8^2\).

Step 3: Simplify the equation

Simplifying the above equation leads to \((x+3)^2 + y^2 = 64\). This is the equation for the given circle.

Key Concepts

Circle Center

In a circle, the center is a crucial component as it defines the exact middle of the circle. The center is represented by a pair of coordinates \(h, k\). These coordinates determine the circle's position in a plane. If you imagine drawing a line from the center to any point on the boundary of the circle, you'll see that all these lines (called radii) are of equal length.

• The center of the circle in our exercise is given as \((-3, 0)\).
• This means the circle is shifted \(-3\) units on the x-axis from the origin, while remaining on the y-axis.
• The center is foundational for forming the equation of the circle, as it helps to determine the exact location of the shape within the coordinate plane.

Understanding the concept of the circle's center assists in placing the circle within the 2D plane accurately. It is the point from which all the measurements are taken uniformly.

Circle Radius

The radius of a circle is another fundamental part, extending from the center to any point on the circle's edge. It directly influences the size of the circle. With a longer radius, the circle appears larger, while a smaller radius results in a smaller circle.

• In this context, our exercise states that the radius is \(8\).
• The radius dictates the size of the circle, and is always a fixed distance in every direction from the center.

When expressing the radius in the general form of a circle equation, it's represented by the \(r\) value, squared as \(r^2\). So, if the radius is \(8\), then \(r^2 = 64\). This squared form is used in the equation to keep everything clear and straightforward.

General Form of Circle Equation

The general form of the equation of a circle is an essential tool for representing circles mathematically. Its typical form is \((x-h)^2 + (y-k)^2 = r^2\). This formula seamlessly integrates both the center and the radius into one equation.

• \(h\) and \(k\) are the x and y coordinates of the circle's center.
• \(r\) represents the radius of the circle.
• The equation shows how any point \(x, y\) on the circle is related to the center and the radius.

In our specific exercise, substituting the values \(h = -3, k = 0,\) and \(r = 8\) gives us \((x + 3)^2 + y^2 = 64\), which represents the precise location and size of the circle in the plane. The clarity of this form makes it a favorite for solving and understanding circle problems in geometry.