Question

If the $X-$coordinate of a point ${}^{\prime }{P}^{\prime }$ on the segment joining $Q(2,2,1)\text{and}R(5,1,-2)$ is $4,$ then $Z-$ coordinate is:

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is A $-1$

$\text{Let}P(4,\beta ,\gamma ),Q(2,2,1)R(5,1,-2)$
Let the point $P,$ divides the segment $QR$ in $\lambda :1$
Using Section formula
$\Rightarrow 4=\frac{5\lambda +2}{\lambda +1}\Rightarrow 4\lambda +4=5\lambda +2\Rightarrow \lambda =2$
Thus, ratio is $2:1$
Therefore $Z$ coordinate will be
$\Rightarrow \gamma =\frac{m{z}_2+n{z}_1}{m+n}$
$\Rightarrow \gamma =\frac{-2\lambda +1}{\lambda +1}=\frac{-2\left(2\right)+1}{2+1}=\frac{-3}{3}$
$\Rightarrow \gamma =-1$