Problem 86 - College Algebra and Calculus: An Applied Approach

Question

Find the standard form of the equation of the specified circle. Center: \((0,0) ;\) radius: 5

Step-by-Step Solution

Answer

The standard form of the equation of the circle is \(x^2 + y^2 = 25\).

Step 1: Identify the formula

The equation of a circle in standard form is given by \(x^2 + y^2 = r^2\), where \((x, y)\) represent the coordinates of any point on the circle and \(r\) is the radius.

Step 2: Apply circle parameters

The given center of the circle is at the origin \((0,0)\), and the radius is 5. Substituting the radius value into the formula, we get: \(x^2 + y^2 = 5^2\).

Step 3: Simplify Equation

Simplify the equation by calculating the square of the radius. This gives: \(x^2 + y^2 = 25\).

Key Concepts

Standard Form

When talking about the circle equations, the standard form is one of the most common ways to write them. This form makes it straightforward to identify different properties of the circle.

In its standard form, the equation of a circle looks like this:

• \((x - h)^2 + (y - k)^2 = r^2\)

In this equation:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

The left side of the equation involves the coordinates \((x, y)\) on the circle, minus the coordinates of the center \((h, k)\). Both squares are added, making the whole expression equal to the square of the radius on the right side.

For instance, if the center is at the origin \((0, 0)\), the equation simplifies to \(x^2 + y^2 = r^2\). This particular case is often simpler to understand and easier to work with.

Radius

The radius of a circle is defined as the distance from the center to any point on its boundary. This is an essential circle parameter and is denoted by the symbol \(r\).

In the standard form of the circle's equation \((x - h)^2 + (y - k)^2 = r^2\), the radius can be identified immediately as the square root of the right-hand side. If \(r = 5\), inserting it into the equation, you get \((x - h)^2 + (y - k)^2 = 25\).

The procedure followed is squaring the radius, such as:

• If \(r = 5\), then \(r^2 = 25\).
• This is inserted into the equation, giving \(x^2 + y^2 = 25\) for a circle centered at \((0,0)\).

The radius essentially dictates the size or distance the circle extends from its center, keeping the circular boundary consistent.

Center of Circle

The center of a circle is a key feature that defines the position of the circle in a coordinate plane. It is denoted by \((h, k)\), which represents the x and y coordinates of the center.

In the standard form equation \((x - h)^2 + (y - k)^2 = r^2\), the terms \((x - h)\) and \((y - k)\) indicate that all points \((x, y)\) on the circle maintain a constant distance from this center.

For a circle centered at the origin \((0,0)\), the equation simplifies to \(x^2 + y^2 = r^2\).

• This means each point on the circle is exactly \(r\) units away from the center point \((0,0)\).

Understanding the center of a circle is crucial for graphing, as it helps define where the circle is actually located in the plane. The center can move the circle horizontally or vertically, depending on its values.