Question

The point of intersection $C$ of the plane $8 x+y+2 z=0$ 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 $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$, then $| a + b + c |$ is equal to ___

Answer 1

The correct answer is 10.
Plane : 8x+y+2z=0
Given line AB:5x−2​=10y−4​=−4z+3​=λ
Any point on line (5λ+2,10λ+4,−4λ−3)
Point of intersection of line and plane 8(5λ+2)+10λ+4−8λ−6=0
λ=−31​
C(31​,32​,−35​)
L:−1x−1​=2y+4​=3z+2​=μ

CD=(−μ+32​)i^+(2μ−314​)j^​+(3μ−31​)k^
(−μ+32​)(−1)+(2μ−314​)2+(3μ−31​)3=0
μ=1411​
CD=42−5​,42−130​,4285​
Direction ratios →(−1,−26,17)
la+b+cl=10