Question

Find the ratio in which the line segment joining the points $(2,4,5)$ and $(3,5,-4)$ is divided by the $xz$-plane.

• A. $4:5$ externally
• B. $2:3$ externally
• C. $1:3$ externally
• D. $4:5$ internally

Answer 1

Answer: (A)

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.