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Q.
Find the ratio in which the line segment joining the points $(2, 4, 5)$ and $(3, 5, -4)$ is divided by the $xz$-plane.
Introduction to Three Dimensional Geometry
Solution:
Let the join of $P(2, 4, 5)$ and $Q(3, 5, -4)$ be divided by $xz$-plane in the ratio $k : 1$ at the point $R(x, y, z)$.
Therefore $x = \frac{3k+2}{k+1}$,
$y=\frac{5k+4}{k+1}$,
$z=\frac{-4k+5}{k+1}$
Since the point $P\left(x,y, z\right)$ lies on $xz$-plane, the $y$-coordinate should be zero,
i.e., $\frac{5k+4}{k+1}=0$
$\Rightarrow k =-\frac{4}{5}$
Hence, the required ratio is $-4 : 5$, i.e., the ratio is $4:5$ externally.