Problem 31 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify given values

The problem states that the center \((h, k) = (0,0)\) and the radius \(r = 7\).

Step 2: Substitution

Now, substitute these values into the standard equation of a circle. \((x-0)^2+(y-0)^2=7^2\)

Step 3: Simplify equation

We can further simplify the equation to its standard form: \(x^2+y^2=49\). This is our final standard equation of the circle.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is a fundamental concept that helps us easily describe a circle on a coordinate plane. The equation is written as: \[(x-h)^2 + (y-k)^2 = r^2\] Here,

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

This form is handy because it explicitly shows both the center and the radius of the circle directly in the formula. By following the expression \((x-h)^2 + (y-k)^2 = r^2\), you can know everything about a circle's position and size. This format allows easy substitution of known quantities to find the equation of a specific circle.

Center of a Circle

The center of a circle is the fixed point from which all points on the circle are equidistant. In the standard form equation of a circle, the center is denoted as \((h, k)\). This point is vital because it gives us the exact location of the circle on the coordinate plane. Knowing the center allows for constructing and visualizing the circle easily. In the problem provided, the circle's center is given as \((0,0)\), which is the origin of the coordinate plane. This means:

• The circle is perfectly centered around the origin.
• This makes calculations straightforward as substituting into the circle's equation negates some terms.

Understanding the center helps in sketching the circle and predicting its interaction with other geometric figures on the plane.

Radius of a Circle

The radius is one of the primary features of a circle, representing the distance from its center to any point on its perimeter. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the term \(r\) represents the radius. Notice that in the equation, we use \(r^2\) — the square of the radius — to form the full expression. In practice, knowing the radius:

• Determines the size of the circle.
• Allows you to calculate other properties like the circumference and area.

For the given problem, the radius is 7, meaning each point on the circle is exactly 7 units away from the center at \((0,0)\). When you substitute this radius into the standard equation, it becomes \(r^2 = 49\), highlighting its significance in forming the circle accurately.