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Question

If a line drawn from the point A(1,2,1) is perpendicular to the line joining P(1,4,6) and Q(5,4,4), then the find the cordinates of the foot of the perpendicular.

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Solution

Let M be the foot of the perpendicular drawn from the point A (1,2,1) to the line joining P (1,4,6) and Q (5,4,4) .
Equation of a line passing through the points (x1,y1,z1)and(x2,y2,z2) is
xx1x2x1=yy1y2y1=zz1z2z1
Equation of the required line passing through P (1, 4, 6) and Q( 5, 4, 4) is
x14=y40=z62=λx=4λ+1;y=4;z=2λ+6
Coordinates of M are (4λ+1,4,2λ+6).........(1)
The direction ratios of AM are
4λ+11,42,2λ+61i.e.4λ,2,2λ+5
The direction ratios of given line are 4,0,-2
Since AM is perpendicular to the given line
4(4λ)+0(2)+(2)(2λ+5)=0λ=12
Putting λ=12 in (i), the coordinates of M are (3,4,5)
Length of perpendicular from A on the given line
AM=(31)2+(42)2+(51)2=24units
The coordinates of M i.e the foot of the perpendicular =(3,4,5)


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