Hyperbola Reflection

{i0,j0},{i1,j1},{n1,m1},{i2,j2}

LE : y^2/b^2 - x^2/a^2 == 1 ;

{x, y} = {a Tan[θ], b Sec[θ]} ;

{i0, j0} = {0, -(a^2 + b^2)^(1/2)} ;

TE : (y y1)/b^2 - (x x1)/a^2 == 1

{i1, j1} = {i1, b/a (a^2 + i1^2)^(1/2)} ;

NE : y = - (a^2y1)/(b^2x1) (x - x1) + y1

{n1, m1} = {((a^2 + b^2) i1)/a^2, 0} ;

{a0,b0}={i0-i1,j0-j1};
{a1,b1}={n1-i1,m1-j1};
{a2,b2}={i2-i1,j2-j1};

α = ArcTan[b0/a0] ;

β = ArcTan[b1/a1] ;

γ = 2β - α = 2 ArcTan[b1/a1] - ArcTan[b0/a0] = ArcTan[b2/a2] ;

y = (j2 - j1)/(i2 - i1) (x - i1) + j1 ;

(j2 - j1)/(i2 - i1) = Tan[2 ArcTan[(m1 - j1)/(n1 - i1)] - ArcTan[(j0 - j1)/(i0 - i1)]] ; y = Tan[2 ArcTan[(m1 - j1)/(n1 - i1)] - ArcTan[(j0 - j1)/(i0 - i1)]] (x - i1) + j1 ;