I want to find the double curve of the projective flat curve

$$ F (x, y, z) = (x ^ 2 + y ^ 2 + z ^ 2) x + t (x ^ 3 + y ^ 3 + z ^ 3) = 0 $$

Where $ (x, y, z) $ is a homogeneous coordinate in the projective 2-room $ mathbb P ^ 2 $. The double curve is the common algebraic equation $ G (u, v, w) = 0 $ under the condition $ F (x, y, z) = 0 $, Where

$$

begin {cases}

u = frac { partial F} { partial x} = 2x ^ 2 + (x ^ 2 + y ^ 2 + z ^ 2) + 3tx ^ 2, \

v = frac { partial F} { partial y} = 2xy ++ 3ty ^ 2, \

w = frac { partial F} { partial z} = 2xz ++ 3tz ^ 2.

end {cases}

$$

So we have to eliminate variables $ x, y, z $ and find algebraic relationship between $ u, v, w $.

I enter Mathematica:

```
Eliminate({(x^2 + y^2 + z^2) x + t (x^3 + y^3 + z^3) == 0,
u == 2 x^2 + (x^2 + y^2 + z^2) + 3 tx^2, v == 2 xy + 3 ty^2,
w == 2 xz + 3 tz^2}, {x, y, z});
```

But the issue is

```
v == 3 ty^2 + 2 xy && w == 3 tz^2 + 2 xz;
```

Note that they exactly match the last two equations from the input, so Mathematica doesn't solve at all! I don't understand why it doesn't eliminate variables $ x, y, z $ as directed. In this post, OP successfully finds the double curve with Eliminate with exactly the same as mine. What is wrong with my method?