How do I find a point at set distances from two other points using the distance formula?

Question

I want to know how to find the coordinates of a point that is a certain distance away from one point, and a different distance from another point.

Example: (I want the method, not the answer to this example.)

What point is 6 units away from (0, 1) and 3 units away from (5,0)?

Answer 1

This one is simple.
I assume you know the distance formula between two points.
D = ((x1-x2)^2 + (y1-y2)^2)^2
Now, we want to find some point (x,y), such that Distance from (x,y) to a known point, say (h,k) is a known distance, z, and distance from (x,y) to another known point, say (a,b) is a known distance c.
1. Let z = distance from (x,y) to (h,k), where (h,k) is a known point
2. Let c = distance from (x,y) to (a,b), where (a,b) is a known point.
3. Write distance formula in full
example:
z = ((x-h)^2 + (y-k)^2))^0.5
c = ((x-a)^2 + (y-b)^2))^0.5
4. We know the specific values of z, c, k, a, and b. For example, we know we might want the distance from (x,y) to (0,1) to be 6 units, hence z =6, h=0, k=1.
Because of this, the next step is to substitue all known values into both equations.
5. We now have two equations, and two variables(x and y). We may solve these equations simulataneously, and hence arrive at our answer for x, and y, which are our x and y coordinates for the desired point respectively.

Answer 2

I would do a graphical analysis of this problem. Graph the two points (0,1) and (5,0) then see what points could be the specified distance from both points.