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Q.
If $A(3,2,-4), B(5,4,-6)$ and $C(9,8,-10)$ are three collinear point then point $C$ divides $AB$
Three Dimensional Geometry
Solution:
Let $C$ divide $A B$ in the ratio $\lambda: 1$. Then
$C \equiv\left(\frac{5 \lambda+3}{\lambda+1}, \frac{4 \lambda+2}{\lambda+1}, \frac{-6 \lambda-4}{\lambda+1}\right)=(9,8-10)$
Comparing the coordinates, we get
$ 5 \lambda+3=9 \lambda+9$ or $ 4 \lambda=-6 $
$\therefore \lambda=-3 / 2$
Also from $4 \lambda+2=8 \lambda+8$ and $-6 \lambda-4=-10 \lambda-10$, we get the same value of $\lambda$.