Solution
1. Find the coordinates of Point C
Point C divides AB in the ratio k:1. Using the section formula:
C=(k+12k−3,k+14k−6,k+1−3k+1)
Since C lies on the plane 8x+y+2z=0:
8(k+12k−3)+(k+14k−6)+2(k+1−3k+1)=0
Substituting k=2 into the coordinates for C:
C=(34−3,38−6,3−6+1)=(31,32,−35)
2. Define the target line
The given line is 11−x=2y+4=3z+2, which we rewrite in standard form:
Any point P on this line is P(1−λ,2λ−4,3λ−2).
3. Find the foot of the perpendicular
The direction ratios (DRs) of the line CP are:
DRs=((1−λ)−31,(2λ−4)−32,(3λ−2)−(−35))
DRs=(32−λ,2λ−314,3λ−31)
Since CP is perpendicular to the line (which has DRs −1,2,3):
−1(32−λ)+2(2λ−314)+3(3λ−31)=0
4. Find the Direction Ratios (a,b,c)
Substitute λ=1411 back into the DRs of CP:
a′=32−1411=4228−33=−425
b′=2(1411)−314=711−314=2133−98=−2165=−42130
c′=3(1411)−31=1433−31=4299−14=4285
The ratios are (−425,−42130,4285). Multiply by −42/5 to get coprime integers:
5. Final Calculation
Final Answer:
The value is 10.
