Okay, let me change then some of your data.Originally Posted byUFOKatarn

--vertex == point.

--edge == circumference.

--Circle K should be: (x-5)^2 +(y-5)^2 = 16 ----(1)

--Circle G: (x-a)^2 +(y-b)^2 = r^2 ----------(2)

--"touch only in one vertex" == "is tangent to"

--mistery == problem only, :-)

If circle G passes through the points (1,1) and (0,2), then,

(1-a)^2 +(1-b)^2 = r^2 ----at (1,1)------(i)

(0-a)^2 +(2-b)^2 = r^2 ----at (0,2)------(ii)

Eliminate r, (i) minus (ii),

(1-a)^2 -(0-a)^2 +(1-b)^2 -(2-b)^2 = 0

(1-a)^2 -(-a)^2 = (2-b)^2 -(1-b)^2

1 -2a +a^2 -a^2 = 4 -4b +b^2 -(1 -2b +b^2)

1 -2a = 3 -2b

1 -2a -3 = -2b

-2a -2 = -2b

b = a+1`-----------**

That means the center of circle G is at point (a,b) == (a,a+1) --------**

Meaning, circle G is

(x -a)^2 +(y -(a+1))^2 = r^2

(x-a)^2 +(y-a-1)^2 = r^2 ---------(2.1)

We still have two unknowns: a and r.

Let us express r in terms of "a".

At point (1,1), using Eq.(2.1),

(1-a)^2 +(1-a-1)^2 = r^2

(1 -2a +a^2) +(a^2) = r^2

2a^2 -2a +1 = r^2 ----------------**

Now, if circle G is tangent to circle K, then the distance between their centers is minimum at the point of tangency. This minimum distance is a straight line joining the centers of the two circles. It is the sum of the radius of K and the radius of G. Thus,

--radius of circle K = sqrt(16) = 4.

--radius of circle G = sqrt(r^2) = sqrt(2a^2 -2a +1).

--center of circle K = (5,5).

--center of cicle G = (a,a+1).

--distance between the two centers = distance between points (5,5) and (a,a+1).

So,

4+r = sqrt[(5-a)^2 +(5 -(a+1))^2]

4+r = sqrt[(5-a)^2 +(5-a-1)^2]

4+r = sqrt[(25 -10a +a^2) +(16 -8a +a^2)]

4+r = sqrt[41 -18a +2a^2]

Plugging in the value of r in terms of "a",

4 +sqrt[2a^2 -2a +1] = sqrt[41 -18a +2a^2]

Rationalize one of the radicals, square both sides,

16 +8sqrt[2a^2 -2a +1] +(2a^2 -2a +1) = 41 -18a +2a^2

Isolate the radical,

8sqrt[2a^2 -2a +1] = 41 -18a +2a^2 -16 -2a^2 +2a -1

8sqrt[2a^2 -2a +1] = 24 -16a

Divide both sides by 8,

sqrt[2a^2 -2a +1] = 3 -2a

Square both sides,

2a^2 -2a +1 = 9 -12a +4a^2

Bring them all to the lefthand side,

2a^2 -2a +1 -9 +12a -4a^2 = 0

-2a^2 +10a -8 = 0

Divide both sides by -2,

a^2 -5a +4 = 0

(a-4)(a-1) = 0

a = 4 or 1 -----------***

When a=4,

b = a+1 = 5

Hence, center of circle G is (a,b) == (4,5)

r = sqrt[2a^2 -2a +1] = sqrt[2(4^2) -2(4) +1] = 5

Therefore, circle G is

(x-4)^2 +(y-5)^2 = 5^2

(x-4)^2 +(y-5)^2 = 25 ---------------------------------answer.

When a=1,

b = a+1 = 2

Hence, center of circle G is (a,b) == (1,2)

r = sqrt[2a^2 -2a +1] = sqrt[2(1^2) -2(1) +1] = 1

Therefore, circle G is

(x-1)^2 +(y-2)^2 = 1^2

(x-1)^2 +(y-2)^2 = 1 ---------------------------------answer.

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I did the (5 -a -1)^2 on paper by "long" multiplication.