Circle 1: center
M_{1} = <0.35,0,0>, radius r_{1}= 0.22.
Circle 2: center on yaxis, tangents Circle 1 from outside in the point S. 
Problem 1:
Circle 2 has the radius r_{2} = 0.30 .
Where on the yaxis (height y_{2}) is the center of the second circle 2 ? (M_{2} = <0,?,0>)
Problem 2:
Circle 2 has the center M_{2} = <0,0.40,0>.
How big must be the radius of a second circle 2, if it intersects (touches) the first circle 1 at the nearest point?
Problem 3:
What are the coordinates of the point S ?
Problem 4:
What are the angles at M_{1} and M_{2} inside the triangle(0,M_{1},M_{2})?

The triangle(O,M1,M2) is rectangular!
Therefore we use the Pythagorean Theorem:
// Problem 1: //
We know two sides of the triangle(O,M1,M2):
d(0,M_{1}) = x_{1} and d(M_{1},M_{2}) = r_{1}+r_{2}.
y_{2} = sqrt( (r_{1}+r_{2})^{2}  x_{1}^{2})
M_{2} = <0,y_{2},0>.
// Problem 2: //
Because of
r_{1}+r_{2} = d(M_{1},M_{2}) = sqrt(x_{1}^{2} + y_{2}^{2}),
we get
d(M_{1},M_{2}) = sqrt( x_{1}^{2}  y_{2}^{2} )
and r_{2} = d(M_{1},M_{2})  r_{1}.
// Problem 3: //
There is a simple proportionality:
x_{S1}/x_{M1} = r_{2}/ (r_{1}+r_{2}) and
y_{S}/y_{M2} = r_{1}/ (r_{1}+r_{2}),
so: x_{S} = x_{M1} · r_{2}/ (r_{1}+r_{2})
y_{S} = y_{M2} · r_{1}/ (r_{1}+r_{2}).
// Problem 4: //
By the inverse trigonometric function i.e. of tan(x):
Angle(M_{1}) = atan ( y_{2}/ x_{1}),
Angle(M_{2}) = 90  Angle(M_{1}).


Tangent circles rendered with POVRay
Note: To avoid any collision with builtin identifiers and reserved words in POVRay,
it's strongly recommanded to use only words beginning with capital letters for all identifiers of variables declared by the user,
i.e. use "R1" instead of "r_{1}"
and use "Y2" instead of "y_{M2}".
#local R1= 0.22;
#local R2= 0.30;
#local M1 = <0.35,0,0>
#local Y2 = sqrt( pow(R1+R2, 2)  pow(M1.x, 2));
#local M2 = <0,Y2,0>; 
Problem 1 in POVRay
#local R1= 0.22;
#local M1 = <0.35,0,0>
#local M2 = <0.40,0,0>
#local R2 = sqrt( pow(M1.x,2)pow(M2.y,2))  R1;

Problem 2 in POVRay
#local XS = M1.x * R2/(R1+R2);
#local YS = M2.y * R1/(R1+R2);
#local S = <XS,YS,0>
#local Angle_M1 = degrees( atan( M2.y / M1.x) );
#local Angle_M2 = 90  Angle_M1; 
Problem 3 + 4 in POVRay
