Problem 56 - Calculus for Biology and Medicine

Question

Find the equation of a circle with center \((2,3)\) and radius \(4 .\)

Step-by-Step Solution

Answer

The equation of the circle is \((x - 2)^2 + (y - 3)^2 = 16\).

Step 1: Understand the Circle Equation Formula

The general formula for the equation of a circle in a Cartesian plane with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). We will use this formula to find the equation of the given circle.

Step 2: Substitute the Center Coordinates

Given that the center of the circle is \((2, 3)\), substitute these values into the formula: \((x - 2)^2 + (y - 3)^2 = r^2\).

Step 3: Substitute the Radius Value

The radius of the circle is given as \(4\). Plug this value into the equation: \((x - 2)^2 + (y - 3)^2 = 4^2\).

Step 4: Simplify the Equation

Simplify the given equation by calculating \(4^2\), which is \(16\). Thus, the equation becomes \((x - 2)^2 + (y - 3)^2 = 16\).

Key Concepts

Cartesian Plane

The Cartesian Plane is a two-dimensional coordinate system that allows us to locate points using an ordered pair of numbers. These numbers are usually written as \((x, y)\), where \(x\) represents the horizontal direction, and \(y\) signifies the vertical direction.
This system is fundamental in coordinate geometry as it provides a framework for graphing shapes, lines, and curves, such as circles.

• The horizontal line is known as the x-axis, while the vertical line is the y-axis.
• The point where these axes intersect is called the origin, represented by the coordinates \((0,0)\).
• The plane is divided into four quadrants, with each quadrant representing a different combination of positive and negative values for \(x\) and \(y\).

Understanding the Cartesian Plane is essential for plotting the equation of a circle or any other geometric shape.

Circle Formula

The circle formula is crucial for expressing the equation of a circle on the Cartesian Plane. To form the equation of a circle with center point \((h, k)\) and radius \(r\), you use the equation:
\[(x - h)^2 + (y - k)^2 = r^2\]
This formula represents all the points \((x, y)\) that lie at a distance \(r\) from the center \((h, k)\).

• The term \((x - h)^2\) represents the square of the horizontal distance from the center.
• The term \((y - k)^2\) stands for the square of the vertical distance from the center.
• The right side \(r^2\) equals the square of the radius—an invariant that ensures all points \((x, y)\) conform to being exactly \(r\) distance away from \((h, k)\).

Let's illustrate with an example: If we have a circle centered at \((2,3)\) with a radius of \(4\), the equation is found by substituting into the formula:
\[(x - 2)^2 + (y - 3)^2 = 4^2\]
And simplifying further gives:
\[(x - 2)^2 + (y - 3)^2 = 16\]

Coordinate Geometry

Coordinate Geometry, or analytical geometry, is a branch of geometry where we use a coordinate system to analyze the properties and relationships of geometric figures.
It provides a bridge between algebra and geometry through graphs and equations.
By utilizing coordinates, we can solve complex geometric problems visually and numerically.

• Coordinate geometry allows for the precise plotting of points, curves, and shapes onto the Cartesian Plane.
• It simplifies the understanding of the geometric properties like distance, angles, and similarity through algebraic representation.
• This method is particularly powerful because it enables the representation of geometric transformations such as translation, rotation, and reflection using algebraic formulas and matrices.

In this realm, the equation of a circle is just one of many instances where coordinate geometry simplifies complex ideas into understandable graph forms.
The exercise we discussed reflects a typical application, teaching us how to apply a formula within this coordinate framework to detail a circle's positioning and size.