A person standing at the junction (crossing) of two straight paths represented by the equations `2x - 3y + 4 = 0` and `3x + 4y -5=0` wants to reach the path whose equation is `6x - 7y + 8 = 0` in the least time. Find equation of the path that he should follow.

Explanation

Equation of given lines are,
`2x-3y+4=0`………`(1)`
`3x+4y-5=0`……….`(2)`
`6x-7y+8=0` ……….`(3)`
Solving equations `(1)` and `(2)` ,
`x=-(1)/(17)` and `y=(22)/(17)`
`:.` Point of intersection `=(-(1)/(17),(22)/(17))`
The person is at this point.
To reach at path line `(3)` in minimum time, he should move perpendicular to line `(3)`.
Let the equation of line perpendicular to line `(3)` is as follows :
`7x+6y=-lambda`.....`(4)`
This line passes through the point `((-1)/(17),(22)/(17))`
`:. (-7)/(17)+(132)/(17)=-lambda`
`implies lambda=(125)/(17)`
From equation `(4)`, `7x+6y=(125)/(17)`
`implies 119x+102y=125`
Which is the required path of the person.