@acer How to select the number a and b so that the system of equations (x+1)^2+(y+3)^2 = 125  and (x-a)^2+(y-b)^2 = 225 have two integral solutions.

@Kitonum How can I use this code in geometry 2 D?

@vv Thank you very much.

@Carl Love Is there a procedure for every equation?

@Kitonum Thank you very much 

@Kitonum I can't get the result with this code.
 

restart;
k := 0;
for a from -5 to 15 do
for b from -5 to 15 do
for c from -5 to 15 do
if igcd(b, c) = 1 and nops({a, b, c}) = 3 then X := [solve(x = a+sqrt(b*x+c))];
if type(X[1], integer) and type(X[2], integer) then k := k+1; L[k] := [a, b, c, X[]] end if end if end do end do end do;
L := convert(L, list);
k;
L;

@Kitonum Thank you very much.

@Kitonum where I add the condition nops({a,b,c,d,e,f})= 6? And some results are not correct. For example
solve(x = (3*x^2+6*x+2)/(x^2+5*x+10), x)

Mathematica out put 37 solutions.

@vv I tried and got the result. Thank you very much.

How Workaround with this sequence?

the function y = (a sin x + b cos x + c)/(d sin x + e cos x + f) (where d^2 + e^2 < f^2) is changed to the form 
A sin (x) + B cos (x) + C = 0.
We know that, the equation has solution when and only when A^2 + B^2 >= C^2. 
restart;
a := 1;
b := 4;
c := 0;
d := 1;
e := -2;
f := 3;
solve((-d*y+a)^2+(-e*y+b)^2 >= (-f*y+c)^2 , y)

@Kitonum I usedWith the function (sin(x)+4*cos(x))/(sin(x)-2*cos(x)+3), 
The order of results Min and Max are not correct.

PS. With the function, (sin(x)-3*cos(x))/(sin(x)+cos(x)+2), the results too bad. 

@acer I got the results. Thank you very much.

I like the function TeXForm[HoldForm]] of Mathematica. It is wonderfull. 



 

@Kitonum If I have a list of some different cubes, how can I use your code?