A) \[\sqrt{21},\,(1,2,8)\]
B) \[3\sqrt{21},\,(3,2,8)\]
C) \[21\sqrt{3},\,(1,2,8)\]
D) \[3\sqrt{21},\,(1,2,8)\]
Correct Answer: D
Solution :
Let M be the foot of perpendicular from \[(7,14,5)\] to the given plane, then PM is normal to the plane. So, its d. r.' s are\[2,4,-1\]. Since PM passes through \[P(7,14,5)\]and has d.r.'s \[2,4,-1\]. |
Therefore, its equation is \[\frac{x-7}{2}=\frac{y-14}{4}=\frac{z-5}{-1}=r,\] |
\[\Rightarrow \,\,x=2r+7,\] \[y=4r+14,\] \[z=-r+5\] |
Co-ordinates of M be \[(2r+7,\,4r+14,\,-r+5)\] |
Since M lies on the plane \[2x+4y-z=2,\]therefore \[2(2r+7)+4(4r+14)-(-r+5)=2\Rightarrow r=-3\] |
Co-ordinates of foot of perpendicular are \[M(1,2,8)\]. |
PM= Length of perpendicular from P \[=\sqrt{{{(1-7)}^{2}}+{{(2-14)}^{2}}+{{(8-5)}^{2}}}=3\sqrt{21}\]. |
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