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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 find the co-ordinates of the foot of the perpendicular.
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
`(x-x_1)/(x_2-x_1) = (y-y_1)/(y_2-y_1)=(z-z_1)/(z_2-z1)`
Equation of the required line passing through P (1, 4, 6) and Q( 5, 4, 4) is
`(x-1)/4=(y-4)/0=(z-6)/-2=lambda`
`x=4lambda+1;y=4;z=-2lambda+6`
∴ Coordinates of M are` (4lambda+1,4,-2lambda+6)` ..........(1)
The direction ratios of AM are
`4lambda+1-1,4-2,-2lambda+6-1`
i.e `4lambda, 2,-2lambda+5`
The direction ratios of given line are 4,0,-2.
Since AM is perpendicular to the given line
`therefore 4(4lambda)+0(2)+(-2)(-2lambda+5)=0`
`therefore lambda=1/2`
Putting ` lambda=1/2` in (i) , the coordinates of M are (3,4,5 ).
Length of perpendicular from A on the given line
`AM=sqrt((3-1)^2+(4-2)^2+(5-1)^2)=sqrt(24)units.`
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