We will use the distance formula to show this result. Recall that having two points with coordinates , then the distance between this two points is given by:

.  We will call the first fixed point . To find the second point, note that the equation of the circle with radius 1 is given by:

. 

Since we are looking for the lower half, we have by solving for y:

( We need only the lower  half in this case , that is why y is .

Hence our second point is .

Using the distance formula we have:. 

We need to simplify this expression a bit:

this gives finally after cancelling and adding:

We will use the distance formula to show this result. Recall that having two points with coordinates , then the distance between these two points is given by:

. 

We will call the first fixed point .

To find the second point, note that the coordinate of the walking person is (x,4) since the x is changing and y is always 4. 

Using the distance formula we have:. 

We need to simplify this expression a bit:

this gives finally after cancelling and adding:

We will use the distance formula to show this result. Recall that having two points with coordinates , then the distance between these two points is given by:

. 

We will call the first fixed point . To find the second point, note that the coordinates of the walking person is (x,0) since James is walking along the x-axis. 

Using the distance formula we have:. 

We need to simplify this expression a bit to get:

To find the center of the circle. We will have to rewrite the expression of the circle.

We have .

Using complete the square method, this gives:

and we know that:

Therefore we have :

is the same as : .

This means that the center is (1,1). Since the moving has (x,4) as coordinates (the y-coordinate is the same, x is changing).

Using the distance formula we have :

.

We can simplify this expression to get :

To find the intersection of the two circles, we will have to solve the simultaneous system:

.

We first write the second part of the system in a simplified manner , we have then:

and this gives:

.

Now we have :

, then this gives  and hence:

and solving for y we have .

We use the fact that and therefore we have:

.

This gives us 

.

Hence

.

Finally we deduce the value of y by plugging in each x value into our original equation and solving for y . When we do this we get , and when .

Now we the points  and .

Using the distance formula we have :

We will have to find these points. After that we will use the distance formula to find the distance between them. To find the intersection we will have to solve the following system:

which we can write by solving for y in the second equation.

Plugging in the value of y in the first system we have:

. This gives:

and hence we have:

.

This gives or  .

We deduce the value of y from:

.

If .

If  

We have the two points of interesection:. Now we use the distance formula to the distance :

We first find the equation of the circle. we have the center is (1,1) and the radius is 4. This gives:

.

We have the equation of the line is y=1.

Now we need to solve the system:

.

Plugging the value of y=1 in the second equation we have:

. Solving for x, we get :

or . Therefore we have the two points:

and . Now we use the distance formula:

Let , then its symmetric with respect to the x-axis is given by:

. Now using the distance formula we have:

Note that  .

Note that if then its symmetric with respect to the y-axis is given by:

.

Now we use the distance formula.

.

Please note that . Therefore the distance is .

Let . We know that its symmetric with respect to the origin is given by:

b=.

Now we will use the distance formula to find the distance between a and b.

We have:

and this gives: .

This shows the required result.