Functions for generating ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$ and plane through point $\vec{p}$ with normal $\vec{n}$, i.e $\vec{n}\cdot(\vec{r}-\vec{p})=0$
el[a_, b_, c_, x_, y_, z_] := x^2/a^2 + y^2/b^2 + z^2/c^2
pl[n_, p_, x_, y_, z_] := z /. First@Solve[n.({x, y, z} - p) == 0, z]
Using as an example: a=1, b=2,c=3 and normal {1,0,par}:
Manipulate[
Column[{Show[
Plot3D[Evaluate[pl[{1, 0, par}, {-1, 0, 0}, x, y, z]], {x, -3,
3}, {y, -3, 3}, Mesh -> None, PlotStyle -> Opacity[0.5]],
RegionPlot3D[
Evaluate[
reg = (el[1, 2, 3, x, y, z] < 1 &&
pl[{1, 0, par}, {-1, 0, 0}, x, y, z] > z)], {x, -3,
3}, {y, -3, 3}, {z, -3, 3}, BoxRatios -> {1, 1, 1},
Mesh -> False, PlotStyle -> Red, PerformanceGoal -> "Quality"],
RegionPlot3D[
Evaluate[el[1, 2, 3, x, y, z] < 0.9], {x, -3, 3}, {y, -3,
3}, {z, -3, 3}, PlotStyle -> Directive[Blue, Opacity[0.2]],
BoxRatios -> {1, 1, 1}, Mesh -> None],
PlotRange -> Table[{-3, 3}, {3}], ImageSize -> 400],
StringForm["Volume of red region:`1`",
NumberForm[RegionMeasure[ImplicitRegion[reg, {x, y, z}]], 3]],
StringForm["Volume of ellipsoid:`1`", NumberForm[N@4 Pi 1 2 3/3, 3]]
}]
, {par, 0.5, 4}]

UPDATE
In relation to comment:
rot[a_, p_, x_, y_, z_] := pl[{Sin[a], 0, Cos[a]}, p, x, y, z]
This interactive graphic shows relation of normal to plane and what I understand is the desired angle from horizontal plane:
Manipulate[
Show[RegionPlot3D[
Evaluate[el[1, 2, 3, x, y, z] < 1], {x, -4, 4}, {y, -4, 4}, {z, -4,
4}, Mesh -> False, PlotStyle -> Opacity[0.3]],
Plot3D[rot[a Degree, {-1, 0, 0}, x, y, z], {x, -4, 4}, {y, -4, 4},
Mesh -> False, PlotStyle -> {Blue, Opacity[0.5]}],
Graphics3D[{{Arrow[2 {{0, 0, 0}, {0, 0, 1}}]}, {Red,
Arrow[2 {{0, 0, 0}, {Sin[a Degree], 0, Cos[a Degree]}}]
},
{Arrow[{{-1, 0, 0}, {1, 0, 0}}]},
{Arrow[{{-1, 0, 0}, {-1 + 2 Cos[a Degree], 0, -2 Sin[a Degree]}}]}
}]], {a, 0, 90, Appearance -> "Labeled"}]
