In what ratio does the xy-plane divide the li | vedclass

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(B) Let the $xy$-plane divide the line segment joining the points $A(1, 2, 3)$ and $B(4, 2, 1)$ in the ratio $\lambda : 1$.The coordinates of the poin

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$3 : 1$ externally

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(B) Let the $xy$-plane divide the line segment joining the points $A(1, 2, 3)$ and $B(4, 2, 1)$ in the ratio $\lambda : 1$.The coordinates of the point of division are given by the section formula:$\left( \frac{4\lambda + 1}{\lambda + 1}, \frac{2\lambda + 2}{\lambda + 1}, \frac{1\lambda + 3}{\lambda + 1} \right)$.Since this point lies on the $xy$-plane,its $z$-coordinate must be $0$.Therefore,$\frac{\lambda + 3}{\lambda + 1} = 0$.Solving for $\lambda$,we get $\lambda + 3 = 0$,which implies $\lambda = -3$.Since the ratio is $\lambda : 1 = -3 : 1$,the negative sign indicates that the division is external.Thus,the $xy$-plane divides the line segment in the ratio $3 : 1$ externally.

In what ratio does the $xy$-plane divide the line segment joining the points $(1, 2, 3)$ and $(4, 2, 1)$?

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