Question
The ratio in which the line joining $$\left( {2,\,4,\,5} \right),\,\left( {3,\,5,\, - 4} \right)$$ is divided by the $$yz$$ plane, is :
A.
$$2:3$$
B.
$$3:2$$
C.
$$ - 2 :3$$
D.
$$4: - 3$$
Answer :
$$2:3$$
Solution :
Let the point $$R$$ divides the line joining the points $$P\left( {2,\,4,\,5} \right)$$ and $$Q\left( {3,\,5,\, - 4} \right)$$ in the ratio $$m : n.$$
$$\therefore $$ The coordinate of $$R$$ is
$$\left( {\frac{{3m - 2n}}{{m + n}},\,\frac{{5m - 4n}}{{m + n}},\,\frac{{ - 4m + 5n}}{{m + n}}} \right)$$
Since, the point $$R$$ is on $$yz$$ -plane, therefore $$x$$-co-ordinate will be zero.
$$\eqalign{ & \therefore \,\frac{{3m - 2n}}{{m + n}} = 0 \cr & \Rightarrow 3m - 2n = 0 \cr & \Rightarrow 3m = 2n \cr & \Rightarrow \frac{m}{n} = \frac{2}{3} \cr} $$
Let the point $$R$$ divides the line joining the points $$P\left( {2,\,4,\,5} \right)$$ and $$Q\left( {3,\,5,\, - 4} \right)$$ in the ratio $$m : n.$$
$$\therefore $$ The coordinate of $$R$$ is
$$\left( {\frac{{3m - 2n}}{{m + n}},\,\frac{{5m - 4n}}{{m + n}},\,\frac{{ - 4m + 5n}}{{m + n}}} \right)$$
Since, the point $$R$$ is on $$yz$$ -plane, therefore $$x$$-co-ordinate will be zero.
$$\eqalign{ & \therefore \,\frac{{3m - 2n}}{{m + n}} = 0 \cr & \Rightarrow 3m - 2n = 0 \cr & \Rightarrow 3m = 2n \cr & \Rightarrow \frac{m}{n} = \frac{2}{3} \cr} $$