Problem 39 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$\text { Center }(-4,0), r=10$$

Step-by-Step Solution

Answer

The standard form of the equation of the given circle is \( (x + 4)^2 + y^2 = 100 \).

Step 1: Identify the given values

The given values are center (h, k) = (-4, 0) and radius 'r' = 10 units.

Step 2: Write the general form of the equation of the circle

The general equation of a circle is \( (x-h)^2 + (y-k)^2 = r^2 \).

Step 3: Substitute the given values into the equation

Now, replace the values of 'h', 'k' and 'r' in the general form of the circle equation. This yields \( (x - (-4))^2 + (y - 0)^2 = 10^2 \).

Key Concepts

Standard Form

The standard form for the equation of a circle is a way to define the geometric shape using a specific algebraic expression. This form helps in understanding the parameters that define the circle, namely its center and radius. The standard form of a circle's equation looks like this:

• \((x-h)^2 + (y-k)^2 = r^2\)

This equation is very similar to the Pythagorean theorem. It states that any point \((x, y)\) on the circle is at a consistent distance from the center \((h, k)\). The distance here is precisely the radius \(r\).
Once you have identified your circle's center \((h, k)\) and its radius \(r\), you can simply plug these values into the standard form to find the equation of your circle. It is a straightforward path from identifying these basic values to the proper circle equation.

Center-Radius Form

The center-radius form is another term for the standard form of the circle's equation, emphasizing the key aspects of the circle: its center and radius. This form starts from the geometric definition of a circle: all points at a fixed distance (the radius) from a central point (the center). The expression \((x-h)^2 + (y-k)^2 = r^2\) captures this definition.
To solve for this form, you need to:

• Identify the center \((h, k)\) from the problem statement; this will be the point which the circle revolves around.
• Determine the radius \(r\), which tells you how wide the circle is.
• Substitute these values into the equation.

For example, with a center at \((-4, 0)\) and radius \(10\), by substituting these into the equation, you'd get \((x - (-4))^2 + (y - 0)^2 = 10^2\). Simplifying gives the circle's equation in its clean center-radius form.

Circle Equation

The equation of a circle is fundamental in both algebra and geometry as it provides a concise way to illustrate a perfect circle. It mathematically represents the set of all points equidistant from a fixed point, the center.

• The center of the circle is the point \((h, k)\),
  meaning equations will adjust depending on changes in these coordinates.
• The radius \(r\), which squared becomes the number on the right side of the equation.
  Usually provided, squaring it is a simple step.

Consider a circle with center \((-4, 0)\) and radius \(10\). Using the standard equation, it forms \((x+4)^2 + (y-0)^2 = 100\). This equation describes every point \((x, y)\) that lies on the edge of the circle, with each point maintaining a distance of 10 from the center. Hence, this circle equation beautifully encapsulates the nature and dimensions of the circle.