r = 1;
listaKatow = Table[RandomReal[{0, 2 Pi}], {i, 100}];
powierzchnia[x_, y_] = 2 x + 3 y^2;
f[x_, y_] = x^2 + y^2 + (2 x + 3 y^2)^2 - r^2;

alfa = listaKatow[[1]];

Result = Solve[{x, y} \[Element]
     InfiniteLine[{{0, 0}, {Cos[alfa], Sin[alfa]}}] &&
    f[x, y] == 0, {x, y}];

xPunktu = Result[[1, 1, 2]];
yPunktu = Result[[1, 2, 2]];
zPunktu = powierzchnia[xPunktu, yPunktu];

xPunktu2 = Result[[2, 1, 2]];
yPunktu2 = Result[[2, 2, 2]];
zPunktu2 = powierzchnia[xPunktu2, yPunktu2];

dlugosc = Sqrt[xPunktu^2 + yPunktu^2 + zPunktu^2]
dlugosc2 = Sqrt[xPunktu2^2 + yPunktu2^2 + zPunktu2^2]

Show[
 ContourPlot[{y*Cos[alfa] - x*Sin[alfa], f[x, y] == 0}, {x, -2.,
   2.}, {y, -2., 2.}],
 Graphics[{{Blue,
    InfiniteLine[{{0, 0}, {Cos[alfa], Sin[alfa]}}]}, {Red,
    Point[{x, y}] /. Result}}]
 ]

Show[
 Plot3D[{2 x + 3 y^2}, {x, -1, 1}, {y, -1, 1}, BoxRatios -> {1, 1, 1}],
 Graphics3D[{Red, PointSize[.04], Point[{0, 0, 0}]}],
 Graphics3D[{Green, PointSize[.04],
   Point[{xPunktu, yPunktu, zPunktu}]}],
 Graphics3D[{Blue, PointSize[.04],
   Point[{xPunktu2, yPunktu2, zPunktu2}]}],
 Graphics3D[{Opacity[0.5], Sphere[{0, 0, 0}, r]}]
 ]