# Distance from a point to a set

• tarheelborn
In summary, to find the distance between (4,2) and the set A2, we use the definition which states that it is the greatest lower bound of the distances between (4,2) and every point in A2. The distance formula is used to calculate the distance between (4,2) and a generic point (x,y) in A2, which is then simplified to D=√(21-8x-4y). To find the minimum distance, the derivative of D is set to 0 and solved for x and y, or the geometry of the situation can be used to find the intersection of the line y=x/2 and the circle x^2+y^2=1."
tarheelborn

## Homework Statement

I need to find $$\rho((4,2), A_2)$$ where $$A_2=\{(x,y)|x^2+y^2=1 \}$$.

## The Attempt at a Solution

I know that the definition says that this distance is the greatest lower bound of $$\{\rho(x,a)|a \in A\}$$. So the distance between $$\rho((4,2), (x,y))= \sqrt{(x-4)^2+(y-2)^2}=\sqrt{x^2+8x+16+y^2-4y+4}= \sqrt{21-8x-4y}$$ and then I am not sure what to do.

tarheelborn said:

## Homework Statement

I need to find $$\rho((4,2), A_2)$$ where $$A_2=\{(x,y)|x^2+y^2=1 \}$$.

## The Attempt at a Solution

I know that the definition says that this distance is the greatest lower bound of $$\{\rho(x,a)|a \in A\}$$. So the distance between $$\rho((4,2), (x,y))= \sqrt{(x-4)^2+(y-2)^2}=\sqrt{x^2+8x+16+y^2-4y+4}= \sqrt{21-8x-4y}$$ and then I am not sure what to do.
Well, that is $D= (x^2+ y^2+ 8x- 4y+ 20)^{1/2}$. Differentiate that and set the derivative equal to 0 to find the minimum.

Or, use geometry! $x^2+ y^2= 1$ is a circle with center at the origin and radius 1. The nearest point to (4, 2) on that circle lies on the straight line from (0, 0) to (4, 2) (an extended radius). The equation of the line through (0, 0) and (4, 2) is y= x/2. Where does that line intersect the circle $x^2+ y^2= 1$

## What is the definition of "distance from a point to a set"?

The distance from a point to a set is a measure of the shortest distance between the point and any point in the set. It represents the minimum distance that the point would have to travel to reach any point in the set.

## How is the distance from a point to a set calculated?

The distance from a point to a set is calculated by finding the distance between the point and all points in the set, and then selecting the smallest distance as the final result.

## What is the significance of the distance from a point to a set in real life?

The distance from a point to a set has many practical applications in real life, such as calculating the minimum distance for a vehicle to travel between two points, determining the closest location for a business or service, and measuring the accuracy of a GPS location.

## What is the difference between distance from a point to a set and distance from a point to a line?

The distance from a point to a set takes into account all points in the set, while the distance from a point to a line only considers the distance from the point to a specific line. Additionally, the distance from a point to a line is calculated using the formula for calculating the distance between a point and a line, while the distance from a point to a set may use a more complex formula depending on the specific set.

## Can the distance from a point to a set be negative?

No, the distance from a point to a set is always a positive value. This is because it represents the shortest distance between the point and the set, and distance is always a positive quantity.

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