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.

Asked by Abhisek | 1 year ago |  144

##### Solution :-

It is given that

2x – 3y + 4 = 0 …… (1)

3x + 4y – 5 = 0 ……. (2)

6x – 7y + 8 = 0 …… (3)

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

By solving equations (1) and (2) we get

x = $$- \dfrac{1}{17}$$ and y = $$\dfrac{22}{17}$$

Hence, the person is standing at point ($$- \dfrac{1}{17}$$,$$\dfrac{22}{17}$$).

We know that the person can reach path (3) in the least time if he walks along the perpendicular line to (3) from point

($$- \dfrac{1}{17}$$,$$\dfrac{22}{17}$$)

Here the slope of the line (3) = $$\dfrac{6}{7}$$

We get the slope of the line perpendicular to line (3)

= $$-\dfrac{1}{\dfrac{6}{7}}= - \dfrac{7}{6}$$

So the equation of line passing through

($$- \dfrac{1}{17}$$,$$\dfrac{22}{17}$$) and having a slope of $$- \dfrac{7}{6}$$ is written as

$$( y- \dfrac{22}{17})= - \dfrac{7}{6}(x+ \dfrac{1}{17})$$

By further calculation

6 (17y – 22) = – 7 (17x + 1)

By multiplication

102y – 132 = – 119x – 7

We get

1119x + 102y = 125

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

Answered by Pragya Singh | 1 year ago

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