Question

$A$ point $R$ with $x$-coordinate $4$ lies on the line segment joining the points $P(2, -3,4)$ and $Q(8,0,10)$. Find the coordinates of the point $R$.

• A. $(1,2,9)$
• B. $(4,2,6)$
• C. $(8,2,5)$
• D. $(4,-2,6)$

Answer 1

Answer: (D)

Let $R(4, y, z)$ be any point which divides the join of $P(2, -3,4)$ and $Q(8,0,10)$ in the ratio $k : 1$ internally.
$\therefore $ Coordinates of $R$ is $\left(\frac{8k+2}{k+1}, \frac{-3}{k+1}, \frac{10k+4}{k+1}\right)$
But $x$ coordinate of $B$ is $4$
So, $\frac{8k+2}{k+1} = 4$
$\Rightarrow 8k+2=4k+4$
$\Rightarrow k=\frac{1}{2}$
Now, $y = \frac{-3}{\frac{1}{2}+1}=\frac{-3}{\frac{3}{2}}=-2$
and $z = \frac{\frac{10\times1}{2}+4}{\frac{1}{2}+1}=\frac{9}{\frac{3}{2}}=6$
Thus, coordinates of $R$ is $\left(4, -2,6\right)$.