Or just use good old algebra and define everything for ParametericPlot3D
(*solve equation for plane, since it is in z direction solve to y*)
v1 = {0.5, -0.5, 1}; v2 = {-0.5, 0.5, 0};(*vectors spanning the plane*)
{a, b, c} = Cross[v1, v2];(*normal vector of plane *)
{x0, y0, z0} = {0, -1, 0.5};(*center of plane*)
plane = a (x- x0) + b (y - y0) + c (z - z0);
planeP = First[{x, y, z} /. Solve[plane == 0, y]];
(*solve equation of sphere, since we need bottom half solve to z, has \
to solutions, top and bottom part*)
{x0, y0, z0} = {0, 0, 0};(*center of sphere*)
r = 6;(*radius of sphere*)
sphere = (x - x0)^2 + (y - y0)^2 + (z - z0)^2 - r^2;
sphereP = First[{x, y, z} /. Solve[sphere == 0, z]];
(*solve the intersection, since it is in z direction solve to x an z, first solution is the bottom half*)
interP = First[{x, y, z} /. Quiet@Solve[{plane == 0., sphere == 0.}, {x, z}]];
(*show the solution*)
p1 = ParametricPlot3D[planeP, {x, -10, 10}, {z, -10, 5}, Mesh -> False, Axes -> False,
RegionFunction -> Function[{x, y, z}, z < 0], PlotStyle -> Gray, Lighting -> "Neutral"];
p2 = ParametricPlot3D[sphereP, {x, -10, 10}, {y, -10, 10}, Mesh -> False, Axes -> False,
PlotStyle -> Gray, Lighting -> "Neutral"];
p3 = ParametricPlot3D[interP, {y, -10, 10}, Mesh -> False, Axes -> False,
PlotStyle -> Directive[{Thick, Red}]];
Show[p1, p2, p3, Mesh -> False, Axes -> False]
