find the angle between the two st lines 3x=4y+7 and 5y=12x+6 and also the equations to the two lines which pass through the point (4,5) and make equal angles with the two given lines

Suppose the lines make angles θ1 and θ2 with the X-axis, and let θ be the angle between them. Using some basic geometry, one can tell that the angle between the lines (θ) will be θ1 - θ2.

Then tanθ = tan(θ1 - θ2), which leads to tanθ = tanθ1−tanθ21+tanθ1tanθ2.

Now, replacing tanθ1 and tanθ2 by m1 and m2 respectively, we get tanθ =

m1−m21+m1m2m1−m21+m1m2
m1−m21+m1m