Find Equation

Question

The distance from a point (x,y) to a point (2,3) is 5. Find the equation that describes this fact.

Answer 1

Let's think about this. We need an equation that relates two points to the distance between them. Luckily that's exactly what the distance formula does, which is:

D = sqrt((X₂ - X₁)² + (Y₂ - Y₁)²))

where D is the distance between points (X₁, Y₁) and (X₂, Y₂)

So, let's take our information and plug it in. We have:

D = 5

(x, y)
(2,3)

 

so, we should write:

5 = √((2 - x)² + (3 - y)²))

What we have is exactly what we wanted: an equation that describes the fact we were given.

Answer 2

The Distance Formula is what we use to find the distance between two points. Here it is:

 

Distance = √[(x₂-x₁)²+(y₂-y₁)²]

 

Let's plug in 5 for the distance, 2 for x₁, x for x₂, 3 for y₁ and y for y₂:

5 = √[(x-2)²+(y-3)²]

 

From here let's simplify by squaring each side:

 

25 = (x-2)²+(y-3)²

 

From here, we can move stuff around to get things in whatever form we want, but this is an equation that describes this situation.