Hi Ibrar,
    As per your formula for 
     x+y-1 = 0.
    We turn this into y = 1 -x.
     y = (-1)x + 1.
    c= 1 here as per y = m*x+c
     $$\frac{\left(y-c\right)}{m}\ =\ -1$$
     For 
     x+y+1 = 0.

    We turn this into y = -1 -x.

    y = (-1)x - 1.

     c= -1 here as per y = m*x+c
    $$\frac{\left(y-c\right)}{m}\ =\ -1$$
So the slope is -1 in both the cases.
The best way to calculate slope is $$-\frac{\left(x\ coefficient\right)}{\left(y\ coeffcient\right)}$$

Hi Blisstry Mystery,
If you read the question carefully, it is given that "The angle between the tangents from point P to the circle with center (3,4) and radius 5 is 120 degrees."
Here, the tangents are PA and PB and angle between tangents is given as 120 degrees. Thus Angle APB is 120 degree.
I hope this helps!