Question:
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.
Answer:
Given equations are:
2x - 3y + 4 = 0 .............1
3x + 4y - 5 = 0..............2
After solving it, we get
x = -1/17, y = 22/17
Suppose that person is standing at point P(-1/17, 22/17)
Again given line is 6x - 7y + 8 = 0
Slope = 6/7
Slope of perpendicular line is = -7/6
Now equation of line passing through (-1/17, 22/17) and perpendicular to 6x - 7y + 8 = 0 is
y - 22/17 = (-7/6)*(x + 1/17)
=> 6(17y - 22)/17 = (-7)*(17x + 1)/17
=> 6(17y - 22) = (-7)*(17x + 1)
=> 102y - 132 = -119x - 7
=> 119x + 102y - 132 + 7 = 0
=> 119x + 102y - 125 = 0
This is the required equation of line.
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