Problem 16 - Algebra 2

Question

Write an equation for the circle that satisfies each set of conditions. center \((0,3),\) radius 7 units

Step-by-Step Solution

Answer

The equation of the circle is \\(x^2 + (y - 3)^2 = 49\\).

Step 1: Understanding Circle Equation

The general equation for a circle with center at \(h, k\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Here, the center \(h, k\) is \(0, 3\) and the radius \(r\) is 7 units.

Step 2: Substitute Center Coordinates

Substitute the center \(h = 0\) and \(k = 3\) into the general equation of a circle. We have: \((x - 0)^2 + (y - 3)^2 = r^2\).

Step 3: Substitute Radius

Substitute the radius \(r = 7\) into the equation to get: \((x - 0)^2 + (y - 3)^2 = 7^2\).

Step 4: Simplify the Equation

Calculate \(7^2\) which is 49. Now, the equation becomes: \(x^2 + (y - 3)^2 = 49\).

Key Concepts

Circle Properties

A circle is a geometric shape that is perfectly round, with each point on its boundary equidistant from a central point. Here are some key properties of a circle that are important to understand:

• The distance from the center to any point on the circle is the same and is known as the radius.
• The circle's center is a fixed point inside the circle, and every point on the circumference is at a radial distance from the center.
• The complete distance around the circle is called the circumference, and it equals \(2\pi r\), where \(r\) is the radius.
• A circle can be described by its equation in a coordinate plane.

This equation is derived from the definition of distance in the Cartesian plane. Understanding these properties makes it easier to derive and manipulate the equation for a circle.

Circle Center

The center of a circle is crucial in defining its equation. Generally denoted as \((h, k)\), this point serves as the anchor from which all radial distances are measured.
In an equation like \((x - h)^2 + (y - k)^2 = r^2\), \((h, k)\) specifies the circle's location on the coordinate plane.

• In our specific problem, the center is located at \((0, 3)\).
• This means that the circle's center is 3 units high on the y-axis, and directly runs through the origin on the x-axis.
• The circle's position changes when the center \((h, k)\) changes, effectively mapping its location.

Locating the circle's center is the first step in forming the equation, as it directs you on how to apply the transformations needed for substituting the center coordinates into the general circle equation.

Radius of a Circle

The radius of a circle is a vital element that significantly influences the circle's size. It measures the length of a line segment from the center of the circle to any point on its boundary. The radius is essential for determining the circle's equation through the term \(r^2\) in the equation \((x - h)^2 + (y - k)^2 = r^2\).

• In the given exercise, the radius is 7 units, implying each point on the circle's boundary is 7 units away from the center \((0, 3)\).
• This radius is used to calculate \(r^2\), which becomes a part of the equation where \(r^2 = 49\).
• A larger radius results in a bigger circle, while a smaller radius produces a smaller circle.

By correctly substituting the radius into the circle equation, one can accurately represent the size and scale of the circle graphically.