Q.

The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the points A(-3, -6, 1) and B (2,  4, -3) divides the line segment AB internally in the ratio k : 1. If  a, b, c (|a|,|b|,|c| are coprime ) are the direction ratios of the perpendicular from the point C on the line 1x1=y+42=z+23, then |a+b+c| is equal to _____

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answer is 10.

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Detailed Solution

Equation of line AB is x+35=y+610=z14=λ
Let point ‘C’ be (5λ3,10λ6,4λ+1)
As ‘C’ lies on 8x+y+2z=0
8(5λ3)+(10λ6)+2(4λ+1)=0λ=23C co-ordinates are 13,23,53
Now,
 x11=y+42=z+23=μ is ' L 
 point on ' L ' is (μ+1,2μ4,3μ2)

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The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the points A(-3, -6, 1) and B (2,  4, -3) divides the line segment AB internally in the ratio k : 1. If  a, b, c (|a|,|b|,|c| are coprime ) are the direction ratios of the perpendicular from the point C on the line 1−x1=y+42=z+23, then |a+b+c| is equal to _____