Problem 31 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$\text { Center }(0,0), r=7$$

Step-by-Step Solution

Answer

The equation of the circle in standard form with the given center (0,0) and radius 7 is \(x^2 + y^2 = 49\)

Step 1: Identify the center and radius

The center is (0,0) and the radius r is 7.

Step 2: Substitute the values in the formula

Substitute these values into the standard form equation. Here, h is 0, k is 0 and r is 7. The equation becomes: \( (x - 0)^2 + (y - 0)^2 = 7^2 \).

Step 3: Simplify the equation

By simplifying the equation, we get the standard form of the equation of the circle: \(x^2 + y^2 = 49\).

Key Concepts

Standard Form

The standard form of the equation of a circle is a simplified way to express the relationship between the x and y coordinates of any point on the circle. It is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, \((h, k)\) are the coordinates of the center of the circle, and \(r\) is the radius. The standard form makes it easy to visualize the circle's properties directly from the equation.

• The terms \((x - h)^2\) and \((y - k)^2\) ensure that the equation accounts for any shift of the circle’s center from the origin.
• \(r^2\) represents the area a point can be from the center to still lie on the circle.

Understanding the standard form helps in quickly deducing a circle's structural elements and identifying the shape described by a specific equation.

Circle Equations

Circle equations come in several forms, but the standard form is the most straightforward for basic analysis. The other main form is the general form of a circle's equation:
\[ x^2 + y^2 + Dx + Ey + F = 0 \]
Here, \(D\), \(E\), and \(F\) are constants that can help determine the circle's center and radius if rearranged into the standard form.

• To convert from the general form to the standard form, you would need to complete the square for both \(x\) and \(y\) terms.
• This conversion makes it easier to read the circle's center and radius directly.

Circle equations are fundamental in geometry and can describe not just perfect circles, but also ellipses, depending on their form and coefficients.

Radius

The radius of a circle is the constant distance from the circle's center to any point on its circumference. In the equation, it appears as \(r\), and in the standard form, its square appears as \(r^2\).

• In the example given, where the radius is 7, it means that every point on the circle is exactly 7 units away from the center \((0,0)\).
• This distance is always positive, and knowing the radius is crucial for constructing the circle or understanding the equation of it.

The radius helps in determining not only the size of the circle but also its other properties, such as area \((πr^2)\) and circumference \((2πr)\).

Center of a Circle

The center of a circle is a point defined by the coordinates \((h, k)\) in the standard form equation. This point indicates where the circle is perfectly balanced in all directions on a coordinate plane.

• For the provided exercise, the center is at the origin, \((0, 0)\). This means the circle is symmetrically placed with respect to both the x and y axes.
• When the center is not at the origin, the equation's terms \((x-h)\) and \((y-k)\) shift the circle accordingly on the plane.

The center is essential for pinpointing the circle's location on a graph and plays a critical role in translating and rotating geometric shapes.