Problem 73 - College Algebra with Modeling and Visualization

Question

Find the standard equation of a circle that satisfies the conditions. Radius \(7,\) center \((3,0) \quad\) 74. Radius \(1,\) center \((0,0)\)

Step-by-Step Solution

Answer

First circle: \((x-3)^2 + y^2 = 49\); second circle: \(x^2 + y^2 = 1\).

Step 1: Understanding the Circle Equation

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

Step 2: Identify Circle Parameters

For the first circle: center \((h, k) = (3, 0)\) and radius \(r = 7\). For the second circle: center \((h, k) = (0, 0)\) and radius \(r = 1\).

Step 3: Substitute for First Circle

Using the circle formula, substitute the center \((3, 0)\) and radius \(7\) into \((x-h)^2 + (y-k)^2 = r^2\). This gives \((x-3)^2 + (y-0)^2 = 7^2\).

Step 4: Simplify First Circle Equation

Simplify the equation: \((x-3)^2 + y^2 = 49\). This represents the first circle's equation.

Step 5: Substitute for Second Circle

Using the circle formula, substitute the center \((0, 0)\) and radius \(1\) into \((x-h)^2 + (y-k)^2 = r^2\). This gives \((x-0)^2 + (y-0)^2 = 1^2\).

Step 6: Simplify Second Circle Equation

Simplify the equation: \(x^2 + y^2 = 1\). This represents the second circle's equation.

Key Concepts

Radius of a Circle

The radius of a circle is a crucial measure in geometry. It defines the size of the circle and is the distance from the center of the circle to any point on its edge.
Understanding how to work with the radius is essential for solving many geometric problems involving circles.

• The radius is always a positive number.
• It can be denoted as "\(r\)" in formulas.
• The radius helps in calculating other circle properties, such as circumference and area.

For example, in the circle with center (3,0) and radius 7, the radius is used to determine the equation of the circle in standard form. Similarly, for a circle with center (0,0) and radius 1, the radius is also central to defining the circle's equation.

Circle Center Coordinates

The center of a circle is defined by its coordinates, typically noted as \((h, k)\). This point is the exact middle of the circle, and every point on the circle’s edge is equidistant from this center point.
The coordinates of the center of a circle play a significant role in its equation.

• In the circle equation \((x-h)^2 + (y-k)^2 = r^2\), \(h\) and \(k\) represent the x and y coordinates of the center, respectively.
• Shifting the center changes the circle's position on the coordinate plane but not its size.

For instance, the circle with center (3,0) lies directly along the x-axis, while the circle with center (0,0) is centered at the origin. Identifying these coordinates accurately is vital for writing the standard form equation of a circle.

Standard Form of a Circle Equation

The standard form of a circle equation is \((x-h)^2 + (y-k)^2 = r^2\). This form incorporates both the center coordinates \((h, k)\) and the radius \(r\) to define the unique properties of the circle.
Writing a circle's equation in standard form allows easy identification and graphing of the circle.

• Replacing \((h, k)\) with specific coordinates positions the circle accurately on a coordinate plane.
• Substituting the radius \(r\) determines the circle's size.

For example, when using the coordinates (3,0) and radius 7, the standard form equation simplifies to \((x-3)^2 + y^2 = 49\). Similarly, for a circle with the center at (0,0) and radius 1, the equation becomes \(x^2 + y^2 = 1\). This form facilitates solving problems that involve circles by providing a consistent method to apply in various situations.