The answer process is as follows:
Because the inclination of the straight line x-y+(√2)/2=0 is 45, and the angle greater than 45 is 90, that is, it is perpendicular to the X axis and passes through the point P(- 1, 3), so the equation of the straight line is x=- 1.
2. The equation of the straight line L is (m? -2m-3)x+(3m？ +m- 1) y = 2m-6, and the intercept of l on the x axis is -3.
Then m=-5/3, or m=3. If the slope is 1, m = (1 √ 65)/8.
The answer process is as follows:
Substitute y = 0 and x =-3 into (m? -2m-3)x+(3m？ +m- 1)y=2m-6 to -3m? +6m+9=2m-6，
Is that 3m? -4m- 15=0, or (3m+5)(m-3)=0. So m=-5/3, or m=3.
If the slope is 1, then m? -2m-3=-3m？ -m+ 1, which is 4m? -m-4=0，m=( 1 √65)/8。
3. The maximum distance from point A (2,3) to the straight line L: xcos θ+ysin θ-2 = 0 equals √ 13-2.
The answer process is as follows:
Let |2cosθ+3sinθ-2|/(cos? θ+sin？ θ)=d(d≥0), then |2cosθ+3sinθ-2|=d, if 2cosθ+3sinθ-2≥0, 2cosθ+3sinθ-2=d,
That is 2√( 1-sin? θ)+3sinθ-2=d，
That is 2√( 1-sin? θ)=d+2-3sinθ，
Square the two sides separately.
4( 1-sin？ θ)=9sin？ θ-6(d+2)sinθ+(d+2)？ .
Simplified to 13sin? θ-6(d+2)sinθ+d？ +4d=0，
Because sinθ is a real number, sinθ has a unique solution when d is maximum, so 36(d+2)? -52(d？ +4d)=0，
9(d+2)？ - 13(d？ +4d)=0，4d？ + 16d-36=0，d？ +4d-9 = 0, (d+2+√13) (d+2-√13) = 0, and take d=√ 13-2.