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 Circle's Equation

The general equation of a circle in the Cartesian plane is \[(x - h)^2 + (y - k)^2 = r^2\] where \((h, k)\) represents the center of the circle and \(r\) is the radius.

Step 2: Identify Given Values

From the problem, note the center of the circle is \((2, 3)\) which means \(h = 2\) and \(k = 3\). The radius \(r\) of the circle is 4.

Step 3: Substitute Values into the Circle's Equation

Replace \(h\) with 2, \(k\) with 3, and \(r\) with 4 in the circle's equation:\[(x - 2)^2 + (y - 3)^2 = 4^2\].

Step 4: Simplify the Equation

Compute \(4^2\) to get 16, thus the equation becomes:\[(x - 2)^2 + (y - 3)^2 = 16\].

Key Concepts

Center of a Circle

The center of a circle is an essential part of understanding the equation and geometry of a circle on the Cartesian plane. In mathematics, the center of a circle is a specific point called
7center point from which all points on the circle are equidistant.

The location of the center is crucial because it determines where the circle is placed in the coordinate system. This point is denoted by the coordinates \((h, k)\) in the circle's equation.

If you're given a circle with center \((2,3)\), you know that:

• h = 2: This is the x-coordinate of the center.
• k = 3: This is the y-coordinate of the center.

This pinpointing of the center allows you to use the general circle equation \((x - h)^2 + (y - k)^2 = r^2\) to find the circle's specific equation.

Radius of a Circle

The radius of a circle, often represented as \(r\), is the distance from the center to any point on the circle's boundary. Knowing the length of the radius helps in defining the circle's size.

The equation of a circle in a Cartesian plane utilizes the radius to determine all points that lies at this specific distance from the center:

The role of the radius in the circle's equation is denoted as \(r^2\) in the equation \((x - h)^2 + (y - k)^2 = r^2\).

• If the radius is 4 like in the original exercise, this is what you plug into the equation as \(r\).
• You'll then compute \(r^2\), which means squaring the radius \(4^2 = 16\).

This approach helps in anything from drawing a circle to applying equations in mathematical problems.

Cartesian Plane

The Cartesian plane is the mathematical playground where we plot graphs and shapes, including circles. It is composed of two perpendicular axes:

• The x-axis: runs horizontally.
• The y-axis: runs vertically.

These axes intersect at a point called the origin, \((0, 0)\), dividing the plane into four quadrants.

Coordinates on this plane are written as ordered pairs \((x, y)\), detailing a point's exact location. When positioned within this plane, the equation of a circle, \((x-h)^2 + (y-k)^2 = r^2\), uses these coordinates.

Understanding the Cartesian plane is important because

• It helps in visualizing where shapes and points are located.
• It gives a frame of reference for solving geometric problems.

In the original exercise, the Cartesian plane allowed us to plot a circle with center \((2,3)\) and a radius of 4, utilizing the entire concept behind circle equations.