f[distance_, A_, B_, a_] :=
Module[{dtemp = 0, point0, vec, norm, u0, t0, curdis = 0, points},
vec = { 1, 2};
point0 = {A, B};
points = {{A, B}};
dtemp = 0;
res = FindInstance[((
point0[[1]] + vec[[1]]*t == u^2/(2 a) - a/2 &&
vec[[1]] <
0) || (point0[[1]] + vec[[1]]*t == - (u^2/(2 a)) + a/2 &&
vec[[1]] > 0)) && point0[[2]] + vec[[2]]*t == u && t > 0, {t,
u}, 1];
u0 = u /. res[[1]];
If[vec[[1]] < 0, point0 = {N[u0^2/(2 a) - a/2], N[u0]},
point0 = {N[-(u0^2/(2 a)) + a/2], N[u0]}];
points = Join[points, {point0}];
If[vec[[1]] < 0,
norm = {-a/Sqrt[a^2 + u0^2], u0/Sqrt[a^2 + u0^2]},
norm = {a/Sqrt[a^2 + u0^2], u0/Sqrt[a^2 + u0^2]}];

vec = N[vec - 2*norm*vec.norm/norm.norm];
While[dtemp + curdis < distance,
dtemp = dtemp + curdis;
res =
FindInstance[((
point0[[1]] + vec[[1]]*t == u^2/(2 a) - a/2 &&
vec[[1]] <
0) || (point0[[1]] + vec[[1]]*t == - (u^2/(2 a)) + a/2 &&
vec[[1]] > 0)) && point0[[2]] + vec[[2]]*t == u &&
t > 0, {t, u}, 1];
u0 = u /. res[[1]];
If[vec[[1]] <= 0, point0 = {N[u0^2/(2 a) - a/2], N[u0]},
point0 = {N[-(u0^2/(2 a)) + a/2], N[u0]}];
points = Join[points, {point0}];
curdis =
N[Sqrt[(u0^2/(2 a) - a/2 - point0[[1]])^2 + (u0^2 -
point0[[2]])^2]];
If[vec[[1]] < 0,
norm = {-a/Sqrt[a^2 + u0^2], u0/Sqrt[a^2 + u0^2]},
norm = {a/Sqrt[a^2 + u0^2], u0/Sqrt[a^2 + u0^2]}];

vec = N[vec - 2*norm*vec.norm/norm.norm];


point0 = point0 + (distance - dtemp)*vec/Norm[vec];
points = Join[points, {point0}];
]

Show[ContourPlot[{y^2 == 10 (x + 2.5),
y^2 == -10 (x - 2.5)}, {x, -15, 15}, {y, -15, 15}],
Graphics[Point[point0]],
Graphics[Line[points]]
]
]
(*Manipulate[{ f[distance,-1,1,5]},{{distance,1},0,500,ControlType\
\[Rule]Animator,AnimationRate\[Rule]10}]*)
f[15, -1.5, 0, 5]