Problem 72 - Algebra 2

Question

Write an equation of a circle with the given center and radius. center \((0,-5),\) radius 4

Step-by-Step Solution

Answer

The equation of the circle is \(x^2+(y+5)^2=16\).

Step 1: Write Down the Standard Equation

The standard equation of a circle is \((x-a)^2+(y-b)^2=r^2\). This equation represents every point \((x, y)\) that is on the circle.

Step 2: Substitute the Given Coordinates for the Center and the Radius

Substitute \(a=0\), \(b=-5\), and \(r=4\) into the standard equation. After substitution, the equation becomes \((x-0)^2+(y-(-5))^2=4^2\).

Step 3: Simplify the Equation

Simplify the equation to obtain the final equation of the circle. It simplifies to \(x^2+(y+5)^2=16\).

Key Concepts

Standard Equation of a Circle

The standard equation of a circle is a mathematical expression used to represent a circle on a coordinate plane. This equation provides a way to specify any circle by defining its center and radius. The formula is expressed as:\[(x-a)^2 + (y-b)^2 = r^2\]Here:

• (x, y) represents any point on the circle.
• (a, b) denotes the circle's center coordinates.
• r is the radius of the circle.

This equation comes from the Pythagorean Theorem — the distance from the center to any point on the circle is the radius. Breaking down the formula further, \((x-a)^2 + (y-b)^2\) calculates the square of the distance from the center \((a, b)\) to a point \((x, y)\), and it equals the square of the radius \(r^2\). Substituting the coordinates of the center and the value of the radius into this formula can give the specific equation for a particular circle.

Center of a Circle

The center of a circle is a crucial component in defining its equation. The center is a fixed point from which all points on the circle are equidistant, which is the radius.
In the standard circle equation, the center is represented by the coordinates \((a, b)\). This means:

• The value \(a\) is the x-coordinate of the center.
• The value \(b\) is the y-coordinate of the center.

For example, if you have a circle centered at \((0, -5)\), you plug \(a = 0\) and \(b = -5\) into the standard circle equation \((x-a)^2 + (y-b)^2 = r^2\).
Thus, the position of the circle relative to the coordinate plane is defined by its center. Adjusting \(a\) or \(b\) will shift the circle horizontally or vertically, respectively, without changing its size.
Understanding the center helps in graphically representing the circle on coordinate axes and determining how it is positioned.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the circle. It is what defines the size and extent of the circle.
In the circle equation, the radius \(r\) appears as \(r^2\). For instance, if a circle's radius is given to be 4, then \(r^2 = 16\). This value is crucial when inserting into the standard circle equation, \((x-a)^2 + (y-b)^2 = r^2\).
The radius is always a positive number because it represents a distance. Sometimes, students may confuse \(r\) with \(r^2\) during calculations, so it’s important to differentiate between the radius and its squared value. To simplify the equation, it is helpful to remember that

• Increasing the radius enlarges the circle.
• Decreasing the radius shrinks the circle.

Thus, understanding the radius is essential for determining the circle's size and ensuring the correct formulation of its equation.