A person standing at the junction 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 the equation of the path that he should follow.

Given:

A person is standing at the junction of the below lines

2x-3y+4=0 eq.1

3x+4y-5=0 eq.2

Solving eq.1 and eq.2,

x=-1/17 and y=22/17

Therefore, the person is standing at point (-1/17,22/17)

The equation of the path is:

6x-7y+8=0 eq.3

When the person walks perpendicular of the eq.3 from point (-1/17,22/17), the person will reach in the least time

Slope of eq.3 = 6/7

Therefore, the slope of the line perpendicular to the eq.3 = -1/6/7 = -7/6

Therefore, the equation of the line passing through (–1/17,22/17) and with the slope of -7/6 is given as

y-22/17=-7/6(x+1/17)

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

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