Maharashtra State BoardHSC Arts 12th Board Exam
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If the lines (x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5 are at right angle then find the value of k - Mathematics and Statistics

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If the lines

`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`

are at right angle then find the value of k

 
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Solution

Given equations of the line are:

Let `bara` and  `bar b` be vectors in the direction of lines `(x-1)/-3=(y-2)/(2k)=(z-3)/2` and  `(x-1)/(3k)=(y-5)/1=(z-6)/-5 " respectively"`

`therefore bar a=-3hati+2khatj+2hatk and bar b=3khati+hatj-5hatk`

`bar a.bar b=-9k+2k-10=-7k-10`

Given lines are at right angle

`theta=90^@`

`costheta=(bar a.bar b)/(|bara|.|barb|)`

`bara.barb=0`

`-7k-10=0`

`k=-10/7`

Concept: Shortest Distance Between Two Lines
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