Untitled

Symbolical mathematical problem (mathematica)
3/y^4==3/x^4+a/(x+2y)^4

ans:
    3/(r^4 x^4) == 3/x^4 + a/(x + 2 r x)^4

ans: Graph and 4 solutions and no its doesn't depend on `a`.

In[538]:= eq =
 Eliminate[3/y^4 == 3/x^4 + a/(x + 2 y)^4 && r == y/x, {x, y}]

Out[538]= -24 r - 72 r^2 - 96 r^3 + (-45 + a) r^4 + 24 r^5 + 72 r^6 +
  96 r^7 + 48 r^8 == 3

In[539]:= r /. Solve[eq && -1 < a < 1, r, Reals]

Out[539]= {ConditionalExpression[
  Root[-3 - 24 #1 - 72 #1^2 - 96 #1^3 + (-45 + a) #1^4 + 24 #1^5 +
     72 #1^6 + 96 #1^7 + 48 #1^8 &, 1], -1 < a < 0 ||
   0 < a < Root[-184528125 + 267553125 #1 + 11238750 #1^2 +
       110250 #1^3 - 225 #1^4 + #1^5 &, 1] ||
   Root[-184528125 + 267553125 #1 + 11238750 #1^2 + 110250 #1^3 -
       225 #1^4 + #1^5 &, 1] < a < 1],
 ConditionalExpression[
  Root[-3 - 24 #1 - 72 #1^2 - 96 #1^3 + (-45 + a) #1^4 + 24 #1^5 +
     72 #1^6 + 96 #1^7 + 48 #1^8 &, 2], -1 < a < 0 ||
   0 < a < Root[-184528125 + 267553125 #1 + 11238750 #1^2 +
       110250 #1^3 - 225 #1^4 + #1^5 &, 1] ||
   Root[-184528125 + 267553125 #1 + 11238750 #1^2 + 110250 #1^3 -
       225 #1^4 + #1^5 &, 1] < a < 1],
 ConditionalExpression[
  Root[-3 - 24 #1 - 72 #1^2 - 96 #1^3 + (-45 + a) #1^4 + 24 #1^5 +
     72 #1^6 + 96 #1^7 + 48 #1^8 &, 3],
  0 < a < Root[-184528125 + 267553125 #1 + 11238750 #1^2 +
      110250 #1^3 - 225 #1^4 + #1^5 &, 1]],
 ConditionalExpression[
  Root[-3 - 24 #1 - 72 #1^2 - 96 #1^3 + (-45 + a) #1^4 + 24 #1^5 +
     72 #1^6 + 96 #1^7 + 48 #1^8 &, 4],
  0 < a < Root[-184528125 + 267553125 #1 + 11238750 #1^2 +
      110250 #1^3 - 225 #1^4 + #1^5 &, 1]]}

Table[Thread[{a,
    Cases[r /. NSolve[eq, r], r_ /; Im[r] == 0]}], {a, -1, 1,
   0.02}] // ListPlot