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].