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Question

A line with directiion ratios 2, 7, -5 is intercepted between the lines
x53=y71=z+21 and x+33=y32=z64. Find the length intercepted between the given lines.___


A

78

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B
9
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C
85
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D
10
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Solution

The correct option is A

78


The general points on the given lines are respectively
P(5+3t,7t,2+t)and Q(33s,3+2s,6+4s).
Direction numbers of PQ are
<33s53t,3+2s7+t,6+4s+2t>
i.e.<83s3t,4+2s+t,8+4st>
If PQ is the desired line then direction numbers of PQ should be proportional to < 2, 7, -5> therefore,
83s3t2=4+2s+t7=8+4st5
Taking first and second numbers, we get
5621s21t=8+4s+2t25s+23t=48 (i)
taking second and third member, we get
2010s5t=56+28s7t38s2t=36 (ii)
Solving (i) and (ii) for t and s we get
s=1 and t=1. The coordinates of P and Q are respectively
(5+3(1),7(1),21)=(2,8,3)
and (33(1),3+2(1),6+4(1))=(0,1,2)
the said line intersects the given lines in the points (2,8,3) and (0,1,2) respectively.
Length of the line intercepted between the given lines
=|PQ|(02)2+(18)2+(2+3)2=78.


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