The equations of the common tangebts to the two hypebolas x^2/a^2-y^2/b^2=1 and y^2/a^2-x^2/b^2=1

Note that hyperbola 1 is the symmetric of hyperbola 2 with line y= x as axis of symmetry.

in addition , hyperbola 1 is also the symmetric of hyperbola 2 with line y= -x as axis of symmetry.

so the tangent lines to both hyperbolas should be in form y= x+t , y= x-t, y=-x+t, y=-x-t

assume that line y= x+ t tangent to hyperbola 1, the we have

x^2/a^2-(x+t)^2/b^2= 1

or (a^2-b^2)x^2+2.a^2.t.x+a^2.t^2+a^2.b^2= 0

the line tangent to hyperbola 1 require that the determinant of above quadratic equation equal zero.

work out in details give us t= + sqrt(a^2-b^2) or t= -sqrt(a^2-b^2)

and equations of 4 tangent lines to both hyperbolas are

y= x+sqrt(a^2-b^2)

y=x-sqrt(a^2-b^2)

y=-x+sqrt(a^2-b^2)

y=-x-sqrt(a^2-b^2)