Student Question

What is the length of the shortest pipe required to be connected between two rails along the lines 6x + 4y - 1 = 0 and 3x + 2y + 17 = 0

Expert Answers

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The rails are along the lines 6x + 4y - 1 = 0 and 3x + 2y + 17 = 0

6x + 4y - 1 = 0

=> 4y = -6x + 1

=> y = (-3/2)x + 1

3x + 2y + 17 = 0

=> 2y = -3x - 17

=> y = (-3/2)x - 17

As the two rails are along lines with the same slope they are parallel to each other.

The shortest pipe connecting the two would be perpendicular to both of them.

The distance between the two lines is the length of the pipe that is perpendicular to them.

For two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0, the distance between them is `|c2 - c1|/sqrt(a^2 + b^2)`

Substituting the values we have `|17 + 1/2|/sqrt(3^2+ 2^2)`

=> 17.5/sqrt 13

=> 4.85

The length of the shortest pipe joining the two rails is 4.85

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