A person standing at the junction (crossing) of two straight paths represented by the equation 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

Solution:

The equations of the given lines are

2x - 3y + 4 = 0 ....(1)

3x + 4y - 5 = 0 ....(2)

6x - 7y + 8 = 0 ....(3)

The person is standing at the junction of the paths represented by lines (1) and (2).

On solving equations (1) and (2), we obtain x = - 1/17 and y = 22/17

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

The person can reach path (3) in the least time if he walks along the perpendicular line to (3) from point (- 1/17, 22/17)

Now,

Slope of the line (3) = 6/7

Slope of the line perpendicular to line (3) = - 1/(6/7) = - 7/6

The equation of the line passing through (- 1/17, 22/17) and having a slope of - 7/6 is given by

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

⇒ 6 (17y - 22) = - 7 (17x + 1)

⇒ 102y - 132 = - 119x - 7

⇒ 119x + 102y = 125

Hence, the path that the person should follow is 119x + 102y = 125

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NCERT Solutions Class 11 Maths Chapter 10 Exercise ME Question 24

A person standing at the junction (crossing) of two straight paths represented by the equation 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

Summary:

A person standing at the junction (crossing) of two straight paths represented by the equation 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. Then the equation of the path that he should follow is 119x + 102y = 125