    Question

# A line with directiion ratios 2, 7, -5 is intercepted between the lines x−53=y−7−1=z+21 and x+3−3=y−32=z−64. 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,7−t,−2+t)and Q(−3−3s,3+2s,6+4s). Direction numbers of PQ are <−3−3s−5−3t,3+2s−7+t,6+4s+2−t> i.e.<−8−3s−3t,−4+2s+t,8+4s−t> If PQ is the desired line then direction numbers of PQ should be proportional to < 2, 7, -5> therefore, −8−3s−3t2=−4+2s+t7=8+4s−t−5 Taking first and second numbers, we get −56−21s−21t=−8+4s+2t⇒25s+23t=−48 (i) taking second and third member, we get 20−10s−5t=56+28s−7t⇒38s−2t=−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),−2−1)=(2,8,−3) and (−3−3(−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|√(0−2)2+(1−8)2+(2+3)2=√78.  Suggest Corrections  0      Similar questions  Explore more