Problem 14 - Calculus for Biology and Medicine

Question

How close does the circle with radius \(\sqrt{2}\) and center \((2,2)\) come to the origin.

Step-by-Step Solution

Answer

The circle gets as close as \( \sqrt{2} \) units to the origin.

Step 1: Understand the context

You need to determine the closest distance between a point (the origin, (0,0)) and a circle. The circle has a center at coordinate (2,2) and a radius of \( \sqrt{2} \).

Step 2: Calculate distance from the center of the circle to the origin

Use the distance formula to find the distance from the center of the circle (2,2) to the origin (0,0). The formula is \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step 3: Apply the formula

Plug the coordinates into the distance formula: \( D = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \).

Step 4: Subtract the radius of the circle

The closest distance from the circle to a point outside it is the distance from the circle's center to that point minus the radius of the circle. Therefore, subtract the circle's radius (\( \sqrt{2} \)) from the distance just calculated (\( 2\sqrt{2} \)).

Step 5: Calculate the minimum distance

The minimum distance from the circle to the origin is \( 2\sqrt{2} - \sqrt{2} = \sqrt{2} \).

Key Concepts

Distance Formula

The distance formula is an essential tool in geometry and coordinate geometry for calculating the distance between two points in a coordinate plane. This formula is derived from the Pythagorean theorem and is expressed as:\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points you are considering.
• \(D\) represents the distance between these two points.

In our case, to find how close the circle comes to the origin, we calculate the distance from the circle's center \((2,2)\) to the origin \((0,0)\). This is done by substituting these coordinates into the formula:\[D = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\]The result, \(2\sqrt{2}\), gives us the direct line distance from the origin to the center of the circle in this context.

Circle Equation

The equation of a circle is a fundamental concept in coordinate geometry, defining points that are all equidistant from a fixed central point, called the center. For a circle centered at \((h, k)\) with a radius \(r\), the equation is:\[(x - h)^2 + (y - k)^2 = r^2\]In the given problem, the circle has a center at \((2, 2)\) and a radius of \(\sqrt{2}\). Therefore, the equation of this circle looks like:\[(x - 2)^2 + (y - 2)^2 = (\sqrt{2})^2\]This equation simplifies to:\[(x - 2)^2 + (y - 2)^2 = 2\]Understanding the equation of a circle helps to visualize its position in the coordinate plane and how its geometry relates to other points, such as the origin in this scenario.

Coordinate Geometry

Coordinate geometry, or analytical geometry, merges algebra and geometry using a coordinate system, allowing us to solve geometrical problems algebraically. It is a powerful tool for determining distances, slopes, and areas, among other things, using coordinates.The problem posed involves determining how close the circle comes to the origin, a classic application of coordinate geometry. By using the principles of coordinate geometry, we:

• Determine the distance from the origin to the circle's center using the distance formula.
• Relate this distance to the circle's radius to find the minimum approach distance to a given point outside of it—the origin, in this case.

Here, we calculated the distance from the origin to the center of the circle as \(2\sqrt{2}\), then subtracted the circle's radius \(\sqrt{2}\) to find the closest distance of the circle from the origin:\[2\sqrt{2} - \sqrt{2} = \sqrt{2}\]Thus, coordinate geometry facilitates solving this classic problem by connecting algebraic expressions to geometrical interpretations.