Solve simultaneous equations

x？ -Really? = 1

∴ a？ = 1，b？ = 1

∴ c？ =a？ +b？ =2

∴ Right focus f (√ 2,0)

(1) The tilt angle is 90 degrees,

The straight line x=√2, which obviously has two intersections with the right branch of hyperbola.

(2) The slope of the straight line exists

Let the straight line be y=k(x-√2)

Substitute into hyperbolic equation x? -Really? = 1

X? -k？ (x-√2)？ = 1

That is, (1-k? )x？ +2√2k？ x-(2k？ + 1)=0

The right branch of a straight line and a hyperbola has two intersections.

Equation (1-k? )x？ +2√2k？ x-(2k？ +1)=0 has two unequal positive roots.

∴ ① discriminant > 0

That is (2√2k? )? +4( 1-k？ )*(2k？ + 1)>0

∴ 4(k？ +1)>0 holds.

(2) the sum of two roots > 0

∴ (-2√2k？ )/( 1-k？ )& gt0

∴·k？ & gt 1

(3) the product of two roots > 0

∴ -(2k？ + 1)//( 1-k？ )& gt0

∴·k？ & gt 1

∴k & gt； 1 or k

The range of tilt angle is [π/4, π/2]u(π/2, 3π/4).

To sum up, the range of inclination angle is [π/4, 3π/4].