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 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.
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.
√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.
