Question
Ratio in which the $$zx$$ -plane divides the join of $$\left( {1,\,2,\,3} \right)$$ and $$\left( {4,\,2,\,1} \right).$$
A.
$$1 : 1$$ internally
B.
$$1 : 1$$ externally
C.
$$2 : 1$$ internally
D.
$$2 : 1$$ externally
Answer :
$$1 : 1$$ externally
Solution :
Suppose $$zx$$ -plane divides the join of $$\left( {1,\,2,\,3} \right)$$ and $$\left( {4,\,2,\,1} \right)$$ in the ratio $$\lambda :1.$$
Then, the co-ordinates of the point of division are
$$\left( {\frac{{4\lambda + 1}}{{\lambda + 1}},\,\frac{{2\lambda + 2}}{{\lambda + 1}},\,\frac{{\lambda + 3}}{{\lambda + 1}}} \right)$$
This point lies on $$zx$$ -plane.
$$\therefore \,y$$ -coordinate $$ = 0 \Rightarrow \frac{{2\lambda + 2}}{{\lambda + 1}} = 0 \Rightarrow \lambda = - 1$$
Hence, $$zx$$ -plane divides the join of $$\left( {1,\,2,\,3} \right)$$ and $$\left( {4,\,2,\,1} \right)$$ externally in the ratio $$1 : 1.$$
Alternate solution :
We know that the $$zx$$ -plane divides the segment joining $$P\left( {{x_1},\,{y_1},\,{z_1}} \right)$$ and $$Q\left( {{x_2},\,{y_2},\,{z_2}} \right)$$ in the ratio $$ - {y_1}:{y_2}.$$
$$\therefore \,zx$$ -plane divides the join of $$\left( {1,\,2,\,3} \right)$$ and $$\left( {4,\,2,\,1} \right)$$ in the ratio $$ - 2:2{\text{ i}}{\text{.e}}{\text{., }}1:1$$ externally.