Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((-2,5) ;\) radius 4

Step-by-Step Solution

Answer

\((x + 2)^2 + (y - 5)^2 = 16\)

Step 1: Understand Circle Equation Format

The equation of a circle in center-radius form is given by \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) represent the center of the circle and \(r\) is the radius.

Step 2: Substitute Center Coordinates

The center of the circle is given as \((-2, 5)\). Substitute \(h = -2\) and \(k = 5\) into the equation.\[ (x + 2)^2 + (y - 5)^2 = r^2 \]

Step 3: Substitute Radius

The radius of the circle is given as 4. Substitute \(r = 4\) into the equation.\[ (x + 2)^2 + (y - 5)^2 = 4^2 \]

Step 4: Simplify the Equation

Calculate the square of the radius, which is 16: \[ (x + 2)^2 + (y - 5)^2 = 16 \]This is the final center-radius form for the given circle.

Key Concepts

center-radius form

The center-radius form of a circle is a simple yet powerful way to express the equation of a circle in geometry. It is written as:\[ (x - h)^2 + (y - k)^2 = r^2 \]where:

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

This format essentially tells you how far any point on the circle is from its center. The squared terms \( x-h \) and \( y-k \) represent the horizontal and vertical distances from the center \( (h, k) \), while \( r^2 \) gives the distance squared from the center to any point on the circle.

Understanding the center-radius form is critical when you want to derive the circle's equation from its center and radius. This makes it easier to graph the circle or analyze it in further geometric problems.

circle geometry

Circle geometry involves understanding the properties and relations of circles within the plane. A circle is a two-dimensional shape where all points maintain a constant distance, called the radius, from a fixed point known as the center. The center-radius form equation provides a bridge to explore these properties.

When you know the coordinates of the center and the length of the radius, you have everything needed to fully define the circle in a 2D space. Important aspects of circle geometry include:

• The **center**: the fixed point, often labeled as \( (h, k) \).
• The **radius**: the constant distance from the center to any point on the circle.
• The **equation**: it shows how the x and y distances from the center relate to the radius.

Grasping these concepts helps you to visualize and solve many geometric problems involving circles.

coordinate substitution

Coordinate substitution is a key step in forming the specific equation of a circle when given certain parameters, like its center and radius.

To execute this, you substitute the given center coordinates \( h,k \) and the radius \( r \) into the standard center-radius formula:\[ (x - h)^2 + (y - k)^2 = r^2 \]This substitution allows you to customize the general formula to represent the specific circle in question. For example, if the center of a circle is given as \((-2,5)\) and the radius is \(4\), you insert these into the formula to get:

• Replace \( h \) with \(-2\) and \( k \) with \(5\).
• Substitute \( r \) with \(4\).

This leads to the equation:\[ (x + 2)^2 + (y - 5)^2 = 16 \]By doing this, you specify the circle's position and size on the coordinate plane.