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.
The equations of the given lines are
2x-3y+4 = 0 --------------- (1)
3x+4y- 5 = 0 ---------------- (2)
6x – 7y + 8 = 0
The person is standing at the junction of the paths represented by lines (1) and (2).
So, on solving equations (1) and (2), we get,
x = 
and y = 
Then, the person is standing at point 
The person can reach path (3) in the least time if he walks along the perpendicular line to (3) from point 
Thus, slope of the line (3) = 
Thus, Slope of the line perpendicular to line (3) = 
The equation of the line passing through 
and having a slope of 
is given by
⇒ 6(17y-22) = -7(17x+1)
⇒ 102y – 132 = 119x -7
⇒ 119x + 102y = 125
Therefore, the path that the person should follow is 119x + 102y = 125.